It was the conservative/lib dem coalition that brought in same-sex marriage, and it’s widely accepted by all political sides as perfectly acceptable. There are of course still fringe groups (usually religious) that oppose it, but that’s not restricted to one political party or another (well, it’s probably more extreme right wing than tories are at the minute). And for the most part, people ignore these (unless imo, they’ve been sucked into American talking points online) The political parties of the uk tend to be slightly more to the left than the US, and socially more left again. Gays aren’t a thing to vote on, probably the next issue is trans people though.
Consider this from they conference: London Assembly chair ejected during Braverman speech https://www.bbc.co.uk/news/uk-england-london-66993794
Less extreme tories are well aware they’ve got a massively bigoted party that don’t represent the majority of the UK.