Possibly the rules have changed under JAA but there was, I believe, a time when, if engaged in a commercial operation,
based in the UK, you would be given a date by which time you had to have your a/c on the British register, can't remember how long was allowed, I do remember a USA registered B737-200 operating out of LGW for about six months at a time, so maybe just a permit is required.
With judicious swapping of a/c probably get away with it and at the risk of repeating myself, under JAA it is probably all different now, just one big, happy family!