"Is UK generally more prude than the rest of Europe, or is it just a small fraction of vocal prudes pushing this through?"
An interesting question. In the 1970s we thought England was heading towards a Danish style attitude. Then after 1979 the Mary Whitehouse brigade again started to gain control of influential channels.
Under the 1997 Labour government there was a mixed message. Equality for LGBTQI lifestyles - but at the same time Political Correctness became a way to "no platform" anyone who didn't conform to certain approved lines. It was noticeable that the younger generation seemed more enlightened about some human rights - but at the same time expressed apparently unthinking intolerance in other areas.
The current twenty-somethings seem to acknowledge gender and sexual fluidity. Ideas with which they agree with many Baby Boomer grandparents who matured in the 1970s.
My Swedish friends have long complained that the USA's vitriolic criticism of Swedish sex education etc had forced their society into a generally more repressive attitude in recent years.
The same influences have undoubtedly affected UK policies. The drafting of the Sexual Offences Act (2003) preferred to take the USA as "best practice" - and ignored the better outcomes in many of our European neighbours. Should the Tory Party succumb to a leadership coup by its more conservative elements - then we can expect the political influence of USA Christian extremists*** to increase.
***as criticised by Pope Francis recently.