Having been fortified with the correct amount of fine red wine from the old box, I now think experiments could be performed to validate what might appear to be a crazy Random Internet Yahoo theory.
Apparently you can reach behind the dash and feel both ends of the little pipe that sends cold air to the glove box. Start the car, turn the AC on, and after a minute feel around. If you feel a blast of cold air on your hand near the center of the console, near the end of the pipe, try and detect the direction it's flowing towards. On the lower console there are a bunch of push buttons for the various environmental functions. In the center there's this weird little rounded nubbin; that's the cabin air temp sensor. If the cold air you feel is blowing towards the back side of that sensor (or where you imagine that sensor might be from behind), that's evidence that my crazy theory might have merit.
Feel the sensor after driving for a while. Is it colder than the rest of the dash? If it is, supports nutso theory.
When the heat inferno starts, place your finger on the nubbin and hold it there. If my theory is correct (or even if my theory is stupid and it's just a sensor issue), the heat blast should stop as the sensor warms up and reports the cabin is no longer at -40 C. Uke suggests a helper for that so you don't drive distracted. Also, YMMV; if you're cold-blooded this may not work because your finger is no match for Precision Bavarian Air Conditioning Run Amok. Find somebody with a warmer appendage.
Play with A/C settings. if this happens even when manual A/C is selected, that means that the auto system isn't trying to fix a cabin temp/set point discrepancy. It's something else.
Select manual mode, and then adjust it so that the system is blowing mostly through the upper vents. If that air is cool, but you're still getting heat from below, it might indicate that a valve has glitched in the heat exchanger, or an air valve has stuck. Try every combination of settings, preferably with a MINI Minion sitting beside you. If anything doesn't make sense, its a clue. If everything works great in Manual Mode, but as soons as you switch to auto the barbeque starts, then that's another clue that evil software is involved.
I don't think this is a problem with insulation or the exhaust system under the car. If that were the case, then as soon as the car reached operating temperature the heat would be blasting through and warming up the footwell.. But Ms Burning, esq reports this doesn't happen until after an hour of driving. That tells me that this isn't an insulation failure or an overheating issue; it's a software/sensor/physical computing system failing to correct for the failure of one or more components. My WAG is that the glovebox A/C pipe has loosened and is messing up the cabin air temp sensor. Or that the sensor itself is whacked.
If it turns out to be something entirely different, I will eat vegetarian crow and seek solace in my box of fine wine.