Right! now you'r talking.
Get the technician to check the Sensor resistance at terminals
D38 & D39 of the ECM.
@10deg. (cold) = 3.5 to 4 KOhm
@80deg. (hot) = 600 - 650 Ohms
You said it sends a signal...but ECM doesn't respond. (but it does when Ign. Key/Off) So the ECM is not at fault here.
The ECM monitors:
the ECT (engine coolant temperature sensor) C24 & C25
Oil temp. sensor C27 & C28
and the blower motor temp sensor (above)
Any of those may interfere (contradict) with proper operation.
Also have him check this:
@ A7, A8, A9 when Ign.Off must have 12v to earth.
@ D26 when Ign/On must have 12v to earth ( & it probably has

couse it runs
A4, A5, E5 & E6 = Earth ECM terminals
Blower motor signal output @ D4 (black&green wire) This is Earth Out signal, so check that you have 12v when Ign/On on the Fan's 'other' wire. The ECM output is DC negative Pulse Width Modulated. i.e a waveform. String of 'square' pulses to the Fan. The ECM controls the Fan Speed by shortening the pulses. IF some idiot had shorthen the wire to +12v it's good by ECM Output. Best checked with a Osilloscope, but a 12v LED will do too or even a ordinary test light. bcose it's a power output.
AC Clutch Relay signal output @ D29 --- // ----- (also earth (constant negative out when AC on)
May conect it to operate the Fan through a relay. not perfect.
Have you 'read' the car with a code reader. Any known fault codes?
My 2 cents...right there
Edit: Forgot to mention those BMW's are notorious for faults in the bulk head harness connectors.