OK, so the story has come to an end and here is how:
yesterday I could not wait until Friday afternoon, especially given that I need a car for this weekend for a 1,000 km trip to Germany/Austria, so I went to OD and gave them the car, explaining the story about the fuses and the shortcircuit that I discovered.
They thanked me and said that they'll do their best to have the car ready for Friday evening.
So I just let it go and enjoyed a nice evening of workout in the gym and then a round of drinks with friends followed by a dinner accompanied by couple of bottles of Brunello di Montalcino.
This morning, with a slightly sore head, I called BMW to arrange pick-up and, to my surprise, they told me that I can pick up the car this morning!!! I "flew" to OD and there it was, my fully functioning "tractor ", with all its systems working well.
Apparently, tinting guys removed the back glass and then messed up one of the controllers in the boot when reinstalling the glass which has electrical heater in it. The system shortcircuited and fuse 142, which is in the circuit together with iDrive, got blown. The controller has also gone.
BMW decided that, even though it is not a 100% guarantee case, they will make it a guarantee case in this instance since the tinting garage works closely with them on many vehicles. I would like to hear a conversation that will soon happen between BMW and tinting place
So car is well, all systems are functioning, several sunny days (+22 C) ahead - great start of the weekend!