I believe that if the aircraft is an American aircraft (N registered) and the pilot in command holds the appropriate FAA licence to fly and instruct and the instruction is following the US system and is training towards a US licence or rating then;
Provided that the pilot in command flies the aircraft legally there is nothing wrong.
However, it is not possible to train student pilots on such aircraft since the FAA student certificate and ability of an FAA instructor to give solo authorisations is not valid in the UK.
If that was not the case then Airlines of US origin would never be able to train (checkout) pilots on flights between the UK and USA or on flights that overfly the UK.
Where there could be a problem is when the operator of the aircraft is UK based but the owner of the aircraft is (has to be) American.......in that case, the UK "legal person" (company) is hiring an aircraft out in the UK jurisdiction and under UK law and that could cause problems. A far tighter way to operate would be for the US based company to own the aircraft and also open an office in the UK to provide training/ hire out the aircraft.......that would seem to sit better with JAR, EASA (so far) etc.
I believe that Naples tried to open an office at Calais some years ago. Don't know what happened but perhaps they could provide some advice.
Regards,
DFC