The BMW system is much simpler but it would get very confused driving off road . 2008 MY Supercharged
Rimini Red / Jet
four zone climate
remote park heater
and no ugly kid windows.
magnus satis quod turpis satis

I would recommend anyone buying a FF with TPMS system to undo the valve caps and check the state of the valve stems. From memory they cost me £ 75 each + £20 per wheel for a local tyre shop to take the wheels / tyres off and replace them.Mine were all corroded and I had to replace the whole lot. I regularly take the caps off and rub vaseline into the threads.

So, after a lot more investigation I think the problem is solved. If a new sensor(s) is/are fitted or the TPMS ECU (module) is replaced the car must remain stationary for at least 15 mins for the system to sort itself out as the ECU needs to locate the position of the sensors. After that, drive for around 10 mins keeping your speed about 18 mph. I also first deflated the four road tyre to 65% of the pressure they should be, then pumped them up to the correct pressure.

Good luck

The vaseline tip is a good one - I and others have had those valves snap - causing the tyre to completely deflate immediately. Mine happened checking the tyre pressures before setting off on holiday - with the boot loaded to the roof