I am by no means an expert, but I bought a ME F and a Super yesterday, both of which had problems with the shutter/mirror. The Super had a locked up shutter, which was fixed after shaking it quite violently (some of it was a bit rusty, which locked it open) The F first had problems with the shutter not going back to the normal 'after-shot' position*, then the shutter did not fire although the winder mechanism worked** (like your problem, apparently).
* Standard solution is to nudge the shutter curtain into place with a fingernail.
** It's caused by the wind mechanism not being able to perform the final 'lock' in which the shutter is set ready for the firing action. There's no visual sign of this, and barely any audible. Apart from just exercising it (without the winder), the common solution seems to use the wind lever and wind about halfway (1/3 or 2/3 seemed to work too), stop, and then wind it fully. After a while of this, the ME F fired as normal, going from one shutter action every five or six winds to every second and then being fully reliable.
Apologies for the quirky English and good luck. If the above doesn't fit your problem, then please describe the state of the shutter and mirror, which often is a good clue.
Edit:
Repairing the cameras may not be worth it (unless it needs a CLA anyway, of course), but a name that came to my mind is Harrow Technical:
http://harrowtechnical.co.uk/