Surely the simplest way would be for one or more of the OTL current British Overseas Territories to be incorporated into the United Kingdom. Gibraltar would be the first that springs to mind, as it's semi-close, had a semi-hostile country next door and would immediately benefit from entry into the EU under the UK banner.
Belize, Hong Kong & Singapore are all other candidates that spring to mind as they all have no entirely friendly neighbours and are small territories, and HK and Singapore were somewhat western aligned and trade orientated. Malta jumps out as a candidate as well as already mentioned due to the OTL history.
Jamaica, Newfoundland, Cyprus are longer shots, but maybe possible.