Check the battery voltage with an alternative voltmeter or multimeter just to be sure it is doing what the installed voltmeter is saying, but it looks very much like the voltage regulator is staying closed circuit (supplying the field winding) despite being over voltage, when cold, yet not when hot. It's an odd failure, but the symptoms are pretty unambiguous.
Here is a diagram of the alternator system from an excellent website with practical advice on diagnosing the system:
The current flowing through the rotor (or field winding) produces the magnetic field which allows the alternator to make electricity. For the output (battery) voltage to go high, the field (rotor) current must be too high, and since the only place it gets current from is via the regulator, the regulator must be faulty.
Notice that the regulator has 3 terminals. It adjusts the resistance between D+ and Df to control the current through the rotor which in turn controls the voltage at D+ relative to D-. It is a positive switching regulator.
MOST car regulators of similar type and packaging are negative switching regulators. D+ connects directly to the field winding and the regulator is inserted between the field and earth instead of between D+ and the field. On these regulators, the resistance between Df and D- is adjusted to control the current through the rotor which in turn controls the voltage at D+ relative to D-.
I was initially sold the incorrect type by my local autoelec. When I worked out what was wrong, I took it back and explained to the apprentice at the front counter, who just looked lost. The boss understood exactly what I meant immediately, (and as a bonus, now treats me as someone who actually has some understanding of what they are buying!)
If you buy from an autoelec rather than a BMW supplier, make sure that whoever sells you a replacement regulator understands the difference between the two types and sells you the correct one.