You must have a CofA, licence, and medical issued by the same state.
The UK is a state, the USA is a state, the EU acts like a state and doubtless will eventually become one.
None of this cares about your nationality, only the nationality of the licence, medical and CofA.
I can legally fly an N-reg aeroplane around the world on my FAA licence and medical, or a G-reg aeroplane around the world on my CAA licence and medical. Until the end of this year I can fly an F-reg aeroplane around the world on my UK issued EASA licence and medical, but on January 1st I no longer can, for which reason I'm transferring that one to Ireland - but any other EASA member country would be equally suitable.
There are some relaxations, there are also some countries who have issues over *residency* - not the same as nationality. So it's somewhat difficult for an EU *resident* to own an N-reg aeroplane: you can do it, but workarounds are needed. I suspect that the UK will be pretty relaxed going forwards about EASA aeroplanes being based in the UK and flown by EASA pilots, as it has for years been over American aeroplanes and licences - that has been an EU problem, not a British one.
G