EASA is not a state and cannot issue licenses. Your UK Pilots licence is issued in accordance with EASA regulations but is issued by the UK and therefore it remains a UK licence. At some point in the future but depending on how the EU/UK negotiations unravel it will be a simple matter of removing "EASA" from the UK issued licences as is the case with the UK passport.
The current UK passports remain valid although they have the words "European Community" written on them. They will only be updated on reissue following the expiry. All new passports are already being issued without the EU printed on them. It will be perfectly possible for such a simple but pragmatic process to be undertaken for pilot licences in my view.