Have re-checked at the workshop:

They do use the GT1 scanner.

The AT fluid did not have any sign of metal or other dirt

Importantly, before these problems, I have had a leak in the cooling system and the engine was driving on very low water for some time. smoke came out of the engine bay and the car was towed. Could be that the engine overheated slightly but the auth. bmw dealer said nothign was damaged.

The cooling system was replaced incl. hoses and water pump, thermostat.

The mechanic said this overheating could have been related to the transmission failure. is it possible and what parts could be affected by this?