In terms of strategy, what remains of the British will likely be focused on closer landmasses. Furthermore, lacking targets, I doubt many places outside of the larger Cities and Conurbations would get bombed. If parts of DC survived, you can be damn sure that the UK fared better due to the sheer amount of disused infrastructure that remains throughout the country.
In terms of plausibility, the UK would either be tribalistic (high king fits that concept) or a hardened, organised and militaristic faction that houses most of the science and technology left over from the war (kinda like the BoS or Enclave). Ethics wise, like I said, closer targets would be more useful, and the surrounding islands would make exquisite defensive bases, not to mention old Sea Forts or Oil Rigs.
People are always so quick to put Britain down in the event of a catastrophe, but they have no idea about the scope and breadth of our history, only parts of our old Empire and Colonies. This country has been built to defend against multiple invasions over thousands of years. The situation would no doubt wipe out most of the populace, but that's true of the WHOLE WORLD including the United States, as is stated in the Fallout games themselves.
As for survival bunkers and the like, it is assumed that Vault-Tec are the only ones who have made nuclear holocaust proof shelters. This doesn't rule out similar schemes worldwide, nor does it acknowledge the bunkers most countries already have.
How many times has the United States been invaded by an enemy army in the last 200 years? In fact, how many times has it EVER been invaded? Most US Bunkers are for VIPs, not as shoreline defense networks.