LONDON—If you’re a Ugandan living in Gibraltar, you can vote in Thursday’s referendum that will decide the fate of Britain’s European Union membership. The same goes for U.K. residents hailing from Tuvalu, a cluster of tiny Pacific islands some 9,000 miles from the British Isles. Irish citizens living in Scotland—they can vote, too.

John Rhys-Davies, an actor famed for his portrayal of warrior dwarf Gimli in “The Lord of the Rings” movies, was born in England and raised by Welsh parents. For nearly three decades, he has lived...