I believe that model has three bleed screws that need to be opened after changing the coolant. You might have air pockets that keep the thermostat from opening.

I would also check the entire system, including different parts of the radiator with a laser temperature gun.

Inside joke but very funny!

On the 2002 Mini, the power steering fan and the cooling system fan is/was on the same circuit.  Check to see that the power steering fan turns freely by hand - if it is 'jammed' with debris, it will pop the fuse disabling the radiator fans.  This circuit should have been modified thru MINI's technical service bulletin but it is worth a check.

Does the radiator fans turn on when you activate the AC?  It should operate on a low speed and high speed modes.

As dan mentioned, is there any sign of oil in the coolant or coolant in the oil?

Have you checked the compression for carbon build-up on the pistons?

Are you certain the timing is correct?  If the chain is off by a tooth, funny things can happen even if the car 'seems' to run okay.

Have you checked the A/F mixture to see if it is running too lean?

Let us know.
Ken

If the head gasket is leaking combustion gases into the cooling system, it should be apparent in the coolant itself - cloudy? discoloured? etc. You might also see steam or whitish "smoke" from the tailpipe, depending on the nature of a  head-gasket leak.

What actually happens when it overheats? Does it boil and release steam and coolant, or is the gauge reading high? Has anyone measured the temperature with instruments such as a digital point-and-shoot thermometer?

My 2002 mini overheats within 5 minutes now. I've had the thermostat, radiator, radiator fan and water pump replaced by my local mechanic. Now they're all stumped. They think it might have a head gasket leak but aren't sure.
Any other possibilities or suggestions?