Based on your photos, your firing order does seem to be correct. If the mechanic turned the distributor a little and it fired and ran for a few seconds, it suggests to me the timing is off.

So check the dizzy cap and rotor - they both should be replaced at the same time. Check for cracks or broken termnals. Remove any dust and dirt from the inside and outside. Use a small file to clean up the outer edge of the rotor tip. It does not ever tough the cap terminals - the spark jumps the gap.

It appears to have a distributor vacuum advance connected. Initially, when an engine begins to start, there's no vacuum or mechanical spark advance. Once it does run enough, it would build vacuum and the spark would advance (mostly for fuel economy). Mechanical advance does not come into play until the engine revs increase beyond a certain point and then the advance is progressive with increasing rpms to a set maximum. That's the advance curve. As rpms increase, vacuum decreases because the throttle is open. Consequently, as rpms increase, the vacuum advance goes away.

I suspect your timing is advanced too much. It would begin to run until engine vacuum comes up and advances the spark too much and the engine dies.

Try disconnecting the vacuum advance and plugging the tube on the carb side of where you disconnect it so you don't make a new air leak. That in itself won't affect the engine - initial timing should be set with it disconnected anyways.

Try starting the engine. If it won't fire even with starter fluid, turn the distributor counter-clockwise (to the left) a little bit at a time. Once you do get it running, set the timing properly. Depending on your engine the stock initial timing at idle should be somewhere between 8 and 12 degrees before top dead centre, though electronic ignition systems will vary.

If you don't have a timing light, you can do it "old-school": Get the engine running and warmed up. Disconnect the vacuum advance and get it idling smoothly. Gently advance the timing until the rpms begin to drop and/or it runs less smoothly. Turn it back and retard the timing until the same thing happens. Set the timing half way between the two extremes and clamp it down. Reconnect the vacuum advance - if it runs rough, retard the timing to smooth it out.

I'll read about the oil pressure relief valve as I'm not familiar with that 

Ok an update, had a mobile mechanic out to work on a newer car and he looked at the Moke briefly. He verified it's getting good spark, plugs in correct order. He tried rotating the distributor a little as I had been and it ran at one point for about 15 seconds before stalling. So it can run... The question is why is it so inconsistent and why is it stalling? He wondered if it just needs a new distributor cap. Any insight based on this new info? I'm with the car this weekend.

try having someone crank it while you spray starting fluid in the carb.  If you didn't remove the float and needle from the float bowl cap, that's a must for a car sitting.   I'd also suggest that you pull the plugs, and remove the oil pressure relief valve and fill it with oil. Put the car in reverse and push it backwards sucking the oil up to the oil pump. Keep filling that hole with oil till it stops taking any oil.  You want to prime the pump so you have oil pressure before you start it or risk engine damage/bearings.  Put the relief valve back in place with plugs out, crank the engine. It will spin fast with no back pressure on the cylinders  give it a few burst on the key for about 10- 15 cranks. That should get it primed. Put the plugs back in and leads on. If you Pop had a Haynes manual, get it out and use it.  It has great trouble shooting sections in every chapter for each system of the vehicle.  Mini is likely what the manual will say. Same for moke mechanically. If you don't have one, Get the Haynes  mini manual for your year.  and you can convert the car's electrical system from pos to neg or back  if it's got a generator by just removing the battery cables and and hooking up the ground.  Then tapping the hot side of the battery to the starter cable.   Don't bother with this now. Get it running. Clean out the needle oriface in the float bowl cap.

If you got a few puffs out of the tailpipe, then you had spark somewhere. Check the spark plug leads are connected in the right sequence and check the static timing (engine off) is close. The sequence of spark plug wires may be in the correct order (or not). If the order is wrong, yopu might be getting spark in once cylinder at a close time and others may be "lost" at the wrong timing.
With the engine  at top dead centre for cylinder one, the dizzy rotor should be pointing up and to the right and at where one of the dizzy cap terminals is. That one should be connected to spark plug 1. Then check the rest for the right sequence. There should be timing marks on the front end of the engine and a notch in the crankshaft pulley. If not, there are  timing marks on the flywheel. Note the timing mark for cylinder 1 and 4 are the same, so you have to check cylinder 1 is on its compression stroke. Both valves need to be closed.

It is unlikely you'd suffer compression loss from sitting too long. Maybe on 1 cylinder if a valve stuck, but an engine will run on 3 or even 2 cylinders.

Thank you for the response! It doesn't have an air filter on it and I had tried starter fluid a few times. Not a ton of it. It would sorta try to fire and I got some black/gray smoke out the tailpipe but it wasn't consistently firing.

I had checked all of the plugs and gotten a spark. I hadn't checked the coil leads since I figured if the plugs had spark the coil was doing its thing?
But I'll give it another once over to see if there's some corrosion or something causing issues.

So considering all of that... what else should I try now? I feel like it's probably something simple since it was running for my dad not that long ago: would it have developed a serious enough compression issue just from sitting!? I may see if a friend of his can come help me look at it since it isn't a straightforward diagnosis.

I copied and pasted that audio and was able to hear it. Barely a partial fire every now and then.

I have zero knowledge of positive ground.

Here is what I would do:

Try again with starter fluid . Do you have an air filter? Take it off and spray directly into the throat of the carb. Have someone crank the engine while you are spraying. You do not need to drench it. It should start. If trying to start, give it another squirt or two and see if you can get it running.

If not, then go back to ignition. Here is what I would do:

Sometimes people remove all the sparkplugs to do this test so that the engine cranks faster and/or does not heat up the starter or drain the battery as much. (With all sparkplugs out and starter spinning faster is good time to do a compression test one cylinder at a time.

Start by removing one sparkplug wire. Holding the insulated part of the wire while wearing rubber gloves and place the metal clip end ¼ inch away from a heavy metal bolt or unpainted part of the engine or frame. Have someone crank the engine.You should see spark. Do this test in shade or darkish. If done in bright light you may not see the spark. A good one you might hear. (Alternately, you could do this test with the sparkplug out, the wire re-attached, and hold (with rubber gloves) the insulated wire attached to the plug so that the other threaded metal base end of the plug is against the block (NO ¼ inch gap). Do it at an angle so you can see the tip of the plug. This would be a test of both the wire and the plug because the spark would be in the tip of the plug where it is supposed to be) If by either method, there is no spark, then perhaps only that plug wire is bad. Test another one. If good, keep testing until all 4 are tested. If more than one is no spark, then either multiple or all the wires are bad, or more likely something in the distributor is broken (like the rotor. Pull it out and examine it for cracks or breaks) or the ignition points and condensor need adjustment and/or replacement. You mentioned turning the distributor a bit. These tests will still be good to do even if the timing is off a few degrees.

Or, you could start from the other end of the ignition system. I think this is less likely but who knows? Before you go to removing points and condensor you could test the coil. Remove the big secondary wire that goes from the top of the coil to the top of the distributor. (Remove it at the distributor end). Wearing rubber gloves for extra insulation, hold the insulated end of the wire with the metal tip ¼ inch from a heavy metal bolt or unpainted part of the engine or frame. While someone cranks the engine for you, hold that wire to hopefully see strong spark. If you do not see a spark, then your coil may be a problem, OR the hot small primary wire leading to coil is intermittently not hot, OR the other small primary wire (from the points and condensor) is not giving the right signal.

Ok.. I'm finally back with the Moke this weekend. So far I've:

1. Went back to positive ground, everything I've read has indicated it should be as such. Flipped the battery and the fuel pump leads.
2. Checked the dash pot; it was, indeed, out of oil. Topped that off.
3. Opened up the float chamber. Float seems fine, moves freely. There was a bunch of gunk in the bottom so I cleaned that out and reassembled and checked that it stops filling around 5/8ths full.

Now when I try to turn it over I can hear it try to fire. I'm not sure what to make of what I'm hearing right now so here's another recording in the hope you guys can point me in the right direction... https://drive.google.com/file/d/1UhlHw36f3HABe6wASv_b9FyUkfumwdRJ/view?usp=sharing

Positive/Negative ground:

I don't know what you should have. if i recall correctly my Mk1 1967/68 Austin Countryman was positive ground but I had it about 50 years ago. (All I have left of it is one hub cap and an aftermarket 1970's tachometer.)  Just to muddify the waters, some people apparently have converted their cars from positive to negative ground.

Hopefully someone with specific Moke knowledge can confirm what it originally had and and how to tell whether it has been converted. But it is beyond my knowledge and experience.

Dan, Harvey, my sincere thanks to both of you for the detailed information. I will dig into this in depth and let you know how it goes when I get a chance to work on the Moke.

Dan, one thing you said: "BIG CAUTION Your Moke MAY be a positive ground vehicle!" Oh boy... I didn't know this was a possibility. Maybe it is a positive ground? I had removed the battery to clean the terminals. When reinstalling, I checked which lead went to ground and hooked that to the negative terminal. I vaguely remember thinking "was this thing in there backwards when I took it out?" THEN there was the fuel pump wired in reverse (which I guess I could see my dad doing since his eyes weren't great) but he was very knowledgable about Minis. But, obviously the starter is working and I am getting spark so... did I mix up the battery? And, more importantly, did I damage anything in doing so?

the dashpot maintenance is easy. Unfortunately it may not solve your starting problem, but do the easy stuff first.

The black center of the silver color dome on the top of the carb will screw out.

Be gentle, it has a long rod (2-3 inches) with a "piston" at the bottom. Some people only pull the rod part way out and add oil. Others will remove the rod all the way out. Either way, oil must be added. See video link at bottom (a twin carb car but process is same)

This piston dips down into a carb oil reservoir (nothing to do with engine lubrication). Carb oil is only a few CCs and your carb oil may be gone or thickened. By pulling it all the way out you will know. The purpose of the oil is to provide resistance to the carb cylinder moving up and down too quickly around the stationary piston as vacuum changes. The mechanism is connected to a tapered needle that varies the amount of gas that is suctioned into the intake.

The maintenance I do is to unscrew the black top, pull the rod and piston all the way out. It should come out easily, but go slowly in case your old oil has turned into jello. Fill the hole with SU oil (it is 20 W, others use 3 in 1 oil, others use 10-30 or tinker with different viscosities. I would not add thick oil as you may have thick gunky oil in there already.) Carb oil needs refilling rarely. Years in between for me but I only drive 1,000 miles /yr or so.

How much to fill? Do not try to fill oil all the way up to the top of the silver dome! You can look down in there and see the top of the bore where the piston has come out of. You fill to just below the top of that part . Just takes a few CCs If you add too much, no harm done, it will get sucked into the engine

When you replace the rod/piston into the hole, jiggle it around slightly and slowly to help it settle through the oil without bending the rod. As the piston enters the oil there will be a definite and significant resistance. Push gently and slowly and the piston and rod will fall until you can screw in the black top .

As stated above, you do not need to do it this way but just pull the rod out partway and add some oil. That way might be best for you if not done it before. Let us know how you did. Harvey

https://www.youtube.com/watch?v=cdI6S8kYtUo

URL: https://www.google.com/search?q=mini+carb+oil+cc&sca_esv=90e47ce38a807d1a&source=hp&ei=q9vhZZfDMNyw5NoPzdyliAk&iflsig=ANes7DE

Enjoy the videos, but a caution: You may find info that is way too deep for what you need to resolve - Ask the questions here before you tackle anything complex. The right answer may be simpler than you think!

Good photos!

Attached is a marked up copy of your engine bay to help identify what you have.

A - voltage regulator. Works with a generator to regulate charging voltage. (don't tinker with it!)
B - horn
C - crankcase ventilator. You might have another one on the back or on the front (radiator) end of the engine.
D - blanking plate for heater control valve. Mokes don't have heaters! By he angle of the bolts, it suggests you have a 1275 engine
E - starter solenoid switch. when you turn the key to the start position, it energizes a coil inside to close the heavy duty contacts to send full battery amperage through the big wires to the starter.

BIG CAUTION Your Moke MAY be a positive ground vehicle!

F - Distributor: appears to be an after-market version that may contain electronic components or be entirely electronic. It isn't an original one. The spark plug wires are excessively long by the way. Something to tidy up - much later!
G - Ignition coil
H - Generator AKA Dynamo.
J - Carburetor. SU brand HS type. HS means the fuel bowl is external and mounted on the side. You can remove the cap to inspect and clean the inside and make sure the fuel level is right. The bottom of the bowl is connected to the bottom of main jet.
K - Carb dash pot. This is a chamber that regulates fuel flow at all speeds to suit the amount of air the throttle isl etting into the engine. The little black cap unscrews to let you check and add oil to the dashpot damper. The damper slows the rate of movement of the dashpot innards so that it does not react to throttle changes too quickly. We'll get into SU carb function later.
L - Brake master cylinder. Your Moke has a very simple braking system - only one circuit. If you follow the tubing, you will find a device that splits the system for the font and rear brakes. (I'm not sure if it is a proportioning valve or just a splitter.) It appears your Dad may ahve changed it for the newer one with the plastic reservoir.
M - Clutch master cylinder. This appears to be the original one to the car, or at least the original type, commonly referred to as the "bean can" type. Mini and Moke clutches are hydraulically operated.
N - Clutch housing. Your clutch appears to be a "Pre-Verto" type, indicated by the length of the actuating arm. "Verto"clutches were a different, later design which has a shorter arm and the slave cylinder mounted at a different angle.
O - Thermostat housing.

Dan–excellent point! A lot of your "check" suggestions are beyond my skill range at present. I'll look around for some videos or if anyone has a good resource please send it along.

Here are some photos of the engine bay, I thought it might answer some questions I'm not sure how to answer about set up (dash pot!?): https://photos.app.goo.gl/tVi6RW7PSLD25e7FA

To recap... back to the beginning. You said:

1. It cranks fine except that you wondered about the spinny sound. So the starter is working OK because it would try  run on starter fluid. Battery cable connections etc were improved.

2. It would try to run on starter fluid, so that tells us that it is getting spark but not fuel. The timing may be off because you cleaned terminals and checked inside the distributor, disturbing its position in the proess. You hae not changed points, condenser and rotor (or the cap).

3. On advice, you checked the fuel pump and discovered the wire connections were reversed. With the fuel line removed at the carb, you did get an ample supply of fuel. Old fule was removed and fresh put in.

Summary: it cranks OK and it tries to run on starter fluid, but not gas, which is getting to the carb. Timing may be off, but it is close enough to make it run.

What we haven't confirmed is whether the carb is working properly.

1. Check that the linkages are working properly, that the choke mechanism operates through its proper range and that the throttle shaft is rotating smoothly. Either may be stuck  and not closing properly.

2. Check that the fuel bowl is filling properly but not over-filing. The float valve may be stuck closed or open. If closed, no fule wo;; get in. If open, too much fuel will flow and the engine will drown in it. Remove a spark plug and check if it is wet or smells like fuel. heck the smell of the engine oil - does it smell pungent or gassy?

3. Identify what type of carb you have. A typical SU mode or something else?  Is the fuel bowl hung on the side or built in? If a SU type, is there oil in the dashpot damper? Tell us what you find and we can go from there.

4. Don't fiddle with the ignition and timing any further until we get it running. If it fires on starter fluid, it should be close enough. Once it runs, we can talk about tuning it better.

Got it, thank you for the explanation. I did bump the distributor when I took the cap off to check for corrosion and it rotated a little bit but I tried to rotate it back accordingly. I tried it with very small rotations (1/16 of an inch?) in either direction until I found a position where it seemed to be firing better. Prior to that it would've been timed correctly, my dad was pretty knowledgeable. So I guess I'm not sure if it could be seriously off at this point? I tried not to move it too far in either direction but I don't know what too far really is.

Agree with Dan's comments as the most likely explanation. If so, keep trying to start it. A new solenoid/bendix or whole new starter may be needed.

Try repeated doses of starter fluid to try to get it going?

Less likely is pre-ignition firing before your piston reaches TDC and forces it back down. It could be true if there was a hot spark plug or hot carbon buildup but your engine is cold. But it could still be pre-ignition from faulty timing. Points and condensor changed?

said another way: kickback is when you go to start an engine but timing or point gap is set wrong or points are corroded/burned. (either can make the timing too advanced). The spark ignites the fuel mixture at the wrong time so the piston is driven back down the opposite way. On car engines it can damage the starter/teeth as the starter gear is designed to retract once the engine starts. I listened to your audio several times again and don't that it was the issue.

Back to Dan.

Could someone please explain "pre-ignition and kickback"?

Could the ignition behavior I'm experiencing be due to a bad battery? I had charged it but it is older so perhaps I should start there. I also think I'm fighting a lot of corrosion, I also had to clean up the battery leads and replace the terminals, so I'll try to clean some other things off. 

Unfortunately my weekend visiting and working on the Moke is up so I'm stuck on this for a few weeks. Thanks to everyone who responded, at least now I know I've got gas where it needs to be. To be continued...

if kicking out the starter gear before letting go of key, that sounds like pre-ignition and kickback, or timing is way far early.

+1 to what Willie-B wrote.
If your starting system has a separate solenoid, check the connections there too.

If the spinny noise happens before you release the key, it might be that the engine fired enough to make the bendix release while the starter is trying to spin it and it free-wheels. That could be due to weak electrical connections or a weak/dirty bendix (don't lube it!) or a tired/weal solenoid.

Make sure both connection points for the engine ground strap are clean and tight. Also check the connections at the starter and the battery ground for the same.