As I understand it (in the UK, and it was based on the EU), if you are a resident of the UK then you must have a UK license to fly in the UK, even if you have an FAA certificate and an FAA plane.
It isn't that you can't fly FAA planes in the UK with an FAA certificate, but must have an EASA license, but that if you are a resident, then you can't and must, respectively.
Since this was something that came from EASA and only just adopted by the UK right around when Brexit actually happened, I would guess that its that way for the EU.