Setting aside all the "Fallout isn't really Fallout" arguments, the only canon references of the UK in Fallout are with certain characters said to come from there. Tenpenny came over to seek his fortune, and Moriarty's father (he seems to be Irish) did as well.
So we know that some form of transatlantic travel still exists. That could be by ship, but I lean towards airship/zeppelin; that's how the Brotherhood made the journey across the U.S. to DC. Either way, it'd be a very risky trip, meaning that things are unpleasant enough there for at least a few people to risk their lives to leave. Particularly without any assurances whatsoever that there'd actually be someone waiting for them when they arrived.
A few random thoughts.
Being a set of islands, supplies and other goods aren't going to be nearly as plentiful as in the American territories. More conflict over scarce resources.
No Vaults. There may have been some other underground shelters (ala WWII bunkers), but they wouldn't have been Vault Tec (different looks, designs, functions). If we're talking about a still basically socialist prewar society, probably something pretty utilitarian. If you want to take the 1980's Thatcher-Reagan Revolution angle, imagine lots of private shelters instead. There's nothing in the canon stating the type of prewar government the UK had, but given the political shift towards Commonwealths in the U.S., there was probably some sort of seismic political shift in its allies as well. Lots of artistic license room.
Nationalism. Xenophobia among people who idealize the memory of a grand British empire. Roll in royalty, maybe Arthurian myths, maybe even a mythologized set of narratives about the world outside the Isles. We know that the U.S. has basically no long range communication beyond ham radios. The UK would essentially be cut off from the world as well, and would be developing its own versions of events for itself.
You're going to want some sort of rising power to drive the main conflict. Something akin to Enclave, NCR, Brotherhood, etc., but obviously none of those. Perhaps something arising from the prewar British intelligence establishment. There's a great book called "Bodyguard of Lies", by Anthony Brown, that tells the story of British intelligence ops in WWII. That'd be a good inspiration source.