Post-1900, there is only one remotely plausible way, and that is if Britain is so ravaged by war while the U.S. comes off relatively unscathed that they seek to be part of the U.S. because their entire government has been destroyed.
Even then, I can't see it being permanent, but rather something like pre-nationhood West Germany where the Allies occupied it in zones till the area is built up again. I suspect that the British would have a desire to get back on their feet and America, being Britain's friend, would have just as much a desire to see it regain its soveriegnty.