rudestuff is right, it is a violation of the Chicago Convention for a state to fail to recognise a pilot licence issued by the state of aircraft registry, except in respect of its own nationals, on an international flight.
Some EASA states offer a light-weight validation, valid for non-commercial operations on any of 28 days in a calendar year, subject to undertaking an acclimatisation flight with an EASA instructor and lodging an application with the relevant competent authority. No EASA medical, exams, or flight tests are required for this particular validation.
Notwithstanding the conflict with the Chicago Convention, the requirement in the Basic Regulation to hold an appropriate EASA or UK licence, as the case may be, does not apply to most Annex I aircraft.
Otherwise, to the extent that the pilot and aircraft operator are distinct, a UK-based US-registered non-Annex I (Part-21 in the UK) aircraft can be flown on a US pilot certificate, including one based on a foreign licence, in EASAland by any pilot regardless of residence status.