I think the definition of conservatism differs very much country to country.
It probably explains to a large extent why the conversations that we read on here about American politics are so different to the conversations that we read about politics in countries such as the UK. It also perhaps explains why many Americans (especially conservatives) often seem to have misconceptions about British politics.
In the UK since the 1920s the main distinction between political ideologies has been between socialism and economic conservatism, rather than between conservatism and liberalism. In terms of social and moral attitudes, both of the main parties have tended to share the views of the majority of the country, often dividing along lines of class and education. For this reason, politicians of most mainstream parties have often tended to be more progressive in their views than many ordinary voters. The classic example would be the death penalty, which remained popular among the general public long after politicians had voted for its abolition.
Civil partnerships and same-sex marriage do reveal a somewhat higher degree of conservatism among Conservative MPs compared with Labour. Civil partnerships were a Labour policy, but received support from the Liberal Democrats and Scottish, Welsh, and Northern Irish nationalists. The Conservatives took no official position, but the legislation was supported by a majority of Conservative MPs, including the party leader, Michael Howard. Same-sex marriage, on the other hand, was a policy of the Conservative and Liberal Democrat coalition government. The legislation was supported by an overwhelming majority of Labour, Liberal Democrat, and nationalist MPs, but slightly more than half of Conservative MPs voted against the bill. One could perhaps say that this is evidence that the Conservative Party is moving in the same direction as the other parties, but is doing so a little more slowly.
On the subject of homosexuality, it is worth noting that in 1965 Clement Attlee described it as “an evil”. Attlee is praised by his admirers on the left for his support for the partial decriminalisation of male homosexuality, but it must be noted that, like many of his contemporaries, he supported homosexual law reform on pragmatic grounds, not because he was particularly liberal. Indeed, Attlee’s approach was in keeping with the position of both the Anglican and Roman Catholic churches. Indeed, it is noteworthy that Britain’s greatest liberal reformer of the 1960s, Roy Jenkins, ultimately left the Labour party to create first the Social Democrats and later the Liberal Democrats.
I would also mention that in the UK, controversial moral issues have traditionally been addressed by means of private members’ bills and free votes. E.g., the Murder (Abolition of Death Penalty) Act 1965 (private member’s bill introduced by Sydney Silverman, a Labour MP), the Abortion Act 1967 (private member’s bill introduced by David Steel, a Liberal), and the Sexual Offences Act 1967 (private member’s bill introduced by Leo Abse, Labour, and the 8th earl of Arran, a Conservative).