Sorry to hear your without your E93.
I got mine back from the dealer last week after it being off the road for a month with exactly the same problem.
I managed to get a Boxster as a loan car mainly because I said that I wouldn't accept anything unless it was a convertible, after all that's what you've paid them to provide. the gave me a mondeo estate at first and needless to say I blew my top at them.
My advice is to keep hastling BMW Uk call centre. I got them to cover my loan payment for the period the car was off the and am seeking further compensation for the inconvinience that was caused.
There may some good news however, apparently, the wait time for the part is coming down as the backlog is being cleared, but unfortunately, I would expect to be without your car for a few weeks.
E93 325 SE, Sparkling Graphite, Black Dakota, hi-gloss black trim, 230's, Folding Mirrors, C/A, Sat Nav, Sport Seats, Front Seat Heating, prof Hi-Fi