I’m not a Brit (except by birth), but I suspect it might be a combination of three things:
1)The experience of WWII. Britain (like the rest of Europe) was in great danger of actually being overrun by a fascist state. Following WWII, many other European nations were overrun by an absolutist communist state. I get the sense that fascism and other forms of governmental authoritarianism feel much closer at hand in Europe than in North America.
2)A parliamentary political system. Unlike presidential-congressional systems, parliamentary systems allow for multiple parties, and so give much greater power to the political fringes. While the UK has a much more moderate-tending system, with two dominant political parties, than many European countries, it is still much more likely that a fascist or communist party could play a large role in government than it is in the US. The British system also lacks the checks-and-balances safeguards and constitutional guarantees of freedoms that the US has. (There are, of course, many strengths to a parliamentary system and to the British system in particular that aren’t relevant here.)
3)Margaret Thatcher. As mentioned by others, the 1980s were a time when the right wing held enormous and long-lasting political power in Britain. The Conservatives under Thatcher changed the political landscape and the role of government in society in a fairly radical way. It was easy to imagine that the political spectrum had permanently moved to the right and that Labour and liberalism would never recover. (And to some extent they haven’t.)
4)The Troubles. I said three, and I was originally going to roll this into the others, especially number 3, but I think it should be explicated. The government undertook some radical and oppressive steps to counter the terrorist bombings of the IRA. Bombings of civilian targets on UK soil were a regular occurrence. Given the political landscape, it was easy to imagine that a worsening of the attacks would throw the whole thing to hell.