There was no issue with EU membership other than on the fringes of the mainstream political parties and dedicated groups like UKIP which were loud but unpopular. Although I support independence for Scotland, the idea was barely discussed until 2006 when the SNP suddenly surged to victory. Now the issue is the most prominent one in Scotland and has been for years.Yet dozens of US States do explicitly allow abortion, and as far as I'm aware, the Supreme Court has not made any ruling to challenge that state of affairs, nor does there seem to be any serious attempt to do so. Indeed, part of the ruling appears to explicitly be based on the idea that it should be up to the politicians to decide what the laws on abortion should be, which implies respecting the rights of individual states to allow abortion:
"It is time to heed the Constitution and return the issue of abortion to the people’s elected representatives. “The permissibility of abortion, and the limitations, upon it, are to be resolved like most important questions in our democracy: by citizens trying to persuade one another and then voting.” Casey, 505 U.S., at 979 (Scalia, J., concurring in judgment in part and dissenting in part). That is what the Constitution and the rule of law demand."
This is just paranoia. There's practically no significant mainstream political support for ending abortion rights in the UK. Simply not going to happen.
All it takes is a few people to make enough noise, then a few prominant mainstream politicians get involved and suddenly something that isn't an issue becomes one. I'd be worried about the UK starting to make things like abortion a mainstream political issue. Our politics is becoming so polarised and Americanised that we simply can't rule out the chance that fringe groups will become emboldened here.