Okay. So I got the car back from the shop. Sadly, although this place specialises in "vintage" cars, they don't know much about BMWs since their specialist in them has passed away.
Anyway, they have stopped the smoking problem, but I don't have anything on the intrument cluster. They brought me back what looks to me like a cover for something or other. I've searched the internet for the part number that is stamped on the side of the cover, but with no joy. I thought if I could find what the cover was for, I might find out what was burning out.
The number stamped on the side of cover is 1377 390, so if anyone knows what it might be for, please let me know.
Will try to upload a picture.