To put it simply, Britain go boom.
They’re well in range of the USSR’s bomber force and have a number of strategic targets the Ruskies will try and eliminate in their first strike. Those same targets are close to civilian areas and a good number of civvies are fried.
If you want a good scenario of how the U.K. fares in a 60s nuclear exchange, watch the BBC movie The War Game. It’s later in the 60s which means the Soviets can spare some ballistic missiles for the home islands rather than having to use all of them on the US, but such details are somewhat irrelevant with any sort of nuclear strike. The effect is still the same and the movie demonstrates that well, not to mention the task of post-attack recovery.