I have added the below for those of you not having any success with the good tricks suggested above.
My 2003 '53' Cooper S (UK) passenger side window had the same problem; no movement when window switch was pressed. I first thought it to be the fuse, so replaced that. No change; window motor just clicked as I pressed the switch. The motor was probably clicking before I changed the fuse, but I did not notice at that point. I then found some forums on the problem. I tried holding both window switches up for 20 seconds and then down, etc, but nothing changed. I tried disconnecting and reconnecting the battery, but no change. I tried thumping the door around where the window motor is positioned. No change. I then, with extreme reluctance, went to my local BMW Mini dealer to see what they recommended. Of course their recommendation was to open my wallet, bend over and think of Scotland! BMW Mini, instead of deducing pretty quickly what the fault was likely to be from a vast history of experience with this common problem, wanted to hook the car up to their diagnostics PC for a FULL vehicle check; £80 + VAT. So, I’d be near £100 worse off and would only have a confirmation of the fault to show for it. i.e. nothing would be repaired at this point.
BMW Mini would then be asking a further £70 + VAT for a new window motor, or £140 + VAT for the regulator together with 1 to 2 hours labour on top of that for fitting (a further £80 to £160 + VAT). To be fair to BMW Mini, the parts would come with a 2 year warranty, so if like a few other Mini owners have found, the problem persists due to a motor/regulator design fault, I’d at least have a bit of comeback.
Needless to say, I left the dealer to have another crack myself. I took off the door panel. This only required the removal of the three torx screws around the interior door handle + the torx screw under the clear reflector near the door lock/unlock indicator pin. It was then a case of prizing the panel away from the door. Not an easy task, but it does pull straight off, away from the door with a bit of persuasion. Prying tools are probably best, but not having any myself, I tackled it with my bare hands. There are a few areas on the door where you can get good leverage to free the panel clips. Try gripping the panel with both thumbs and forefingers and lever/push against the door with your other fingers.
After the panel was removed, I could see what I was dealing with and that I didn’t want to go pulling things out in case I broke something and then required urgent BMW Mini Dealer attention.....££££££!
I tried the ‘Fonz’ trick again, but the motor did not budge; even holding the switch, starting the car and hitting the motor simultaneously would not work for me. I was about to admit defeat and book the car into BMW Mini, but then gave this last thing a try. I grabbed one of the regulator cables, just at the point before the cable goes around the white (or green in some models, I believe) sheave/pulley at the bottom of the door on the front-most of the two black window guides (i.e. window guide nearest motor). I got a good grip of the cable and tugged it each way as I started the car then pressed the window switch....................hey presto! The window went down. I really don’t know if my regulator or motor are on their way out, so I may have to replace either or both at some point soon. One point I noted was that the window runs up and down a lot smoother than before. Previously the window ran rather juddery as if on a dry mechanism. I am unclear at the moment what my yanking of the regulator cable has actually done for the motor or regulator, but hopefully it all works for a wee while.
I have looked at second hand motors and regulators, which are £40 for the full assembly. Nice idea, but the second hand parts may already be in the same state or worse as my current parts. If I was going to the trouble, I think I’d rather have new parts installed (with a warranty).
Anyway, I hope this helps someone out there and keeps you away from the expensive dealers for a little while at least.
Good luck.