I understand the regulations this way:
FAA license is required (flying an N-Registered aircraft).
But:
FAA also states that a license of the country where the aircraft is operated in might suffice (hence you only need an EASA license, medical and type rating.)
But:
Flying any aircraft within UK airspace, you have to comply with UK CAA requirements (not EASA).
Each CAA has agreements with the FAA (or not) regarding the operation of a foreign registered aircraft in their country... For the UK, this was written down in "Lasors".
I believe the current state is that if you want to operate a N registered aircraft commercially in the UK, you need both licenses-Faa and CAA..
To my knowledge, Germany for example does not have an agreement like this at all...