Was the liberalisation of Britain supported by the public?

Yes. In a democracy like Britain's the law follows public opinion, it does not form it. The divorce laws changed because peoples attitudes to marriage and divorce changed. The abortion laws were changed because too many women were dying from botched back street abortions. The moors murders would have happened regardless of whether the death penalty was in place or not - and it was abolished because mishandled murder trials led to the hanging of innocent people, leading to public outcry.