Alex Massie connects British secularism to Tory support for marriage equality:
It used to be said that the Church of England was the Tory party at prayer. Doubtless it remains the case that religious Britons are, all other matters being equal, more likely to vote Conservative. But as far as civil marriage is concerned, there is no need to pander to the objections of a faithful minority. Indeed, Britain is a largely secular society these days and, at least as far as same-sex relationships are concerned, a much more civilized place than was the case in years gone by.
Which suggests to me that the core reason the GOP is headed in the opposite direction is religion. Not politics, fundamentalist dogma.