I had this happen to me before, I think this was because the screw inside for pusher came loose somehow. In my impatience I opened it up and tightened it myself but thinking back now, I should have taken it to a watchmaker..
So cost - IDK, I guess it depends on if the watchmaker feels that it's a simple fix or not, but shouldn't be too high if any.
Unfortunately I don't know of places for service in London though. Only for vintage speedys.