“US vs USSR, and the UK is the one paying the price.”
Yes, that was always a key part of the Soviet divide and conquer strategy and propaganda targeting the societies of the Western Allies. The Soviets financed and helped to organize the CND and their political protests which you are seeing depicted in the movie. It was no accident when the CND Peace symbol was adopted for the anti-war and anti-nuclear weapons protest movements that focused on disarming the West and not the Soviet and other Communist nations.
I don’t know much about the UK “ban the bomb” movement but what I do know comes from my readings on the British underground/alternative press of the early-mid-60s that was there when Beats (beatniks) and rock and roll came into fashion.
IT (International Times), OZ, the Beats at Royal Albert Hall. Before those foundational events, there were the stirrings of the “ban the bomb” movement.
Even in the 1980s, the UK Left was certain that Reagan was going to bring about a nuclear war.
There are other bits of hysteria that never come up in discussions of “red scare paranoia”. The Left has their own kooky conspiracy phobias.
https://en.wikipedia.org/wiki/A_Very_British_Coup
(Written in 1981, adapted for the telly in 1988).