The following is necessarily a massive oversimplification.
Religiousness
America is, by and large, the most religious of the high-GDP countries that aren't Middle Eastern (and heavily Muslim) oil exporting nations. Compared to most Western European nations especially, the US havehas a significantly higher number of people who consider themselves highly religious, and a notably smaller proportion of those who identify as non-religious. In the UK, some 55% of people say they are not religious, with about 34% identifying as Christian. By contrast, in the US some 63% of people identify as Christian, most of them Protestants, and only 28% identify as non-religious. That's about a 30% swing. Imagine how much different the UK would be if a quarter of its population swapped over to being heavily religious from non-religious.
As such, religious moralism, especially Protestant Christian moralism, plays a major role in all aspects of American society and politics at large. And Protestant (and Evangelical) Christianity largely sees abortion as one of the gravest sins imaginable (or at least, it seems the gravest one they're willing to get riled up about at the moment). There's simply no comparable religious undercurrent in the UK. While there are surely religious people there who take issue with abortion, they do not pose nearly as large of a sector of the voting (and politically vocal) populace.
Ease of change
Add to this the contrast in "reversibility" on the legality of abortion. In the UK, all Parliament has to do is pass a new law, requiring simply a majority of MPs, and boom, you can completely alter if something is allowed or not. You can even override court decisions this way, as Parliament is ultimately the supreme authority. Contrast in America, where a right to abortion has thus far been held as constitutionally protected by our Supreme Court. And the only thing that can override the Supreme Court on this is itself or a constitutional amendment.
An amendment normally requires congressional two-thirds majorities in both chambers (only one of which is proportionate to population, but gerrymandered) and three quarters of states (which is definitely not proportionate to national population). This is obscenely difficult, and in theory a rather small fraction of our population can prevent any amendment, even if everyone else is vigorously in favor. Thus the much simpler solution: alter the Supreme Court to your view point. Which is still hard due to lifetime appointments, but easier than an amendment you don't have massive majorities for these days. And if you succeed, that difficulty now protects your success. The only way to go about that is through national politics, as Justices are appointed and confirmed through the federal political branches.
All told this created a massive political pressure. Abortion, once a state issue with little national character, was catapulted into a primary issue of national importance because a significant fraction of the population abhorred the constitutional protection of abortion, and so had to mobilize in the single plausible direction of correction: a massive alteration of national politics aimed to alter the Supreme Court's jurisprudence. This sort of urgent need to exploit national politics doesn't seem to arise in UK's system because simple majorities in the Commons are all you ever need, and the appointment of judges to their Supreme Court or other courts are significantly less political—at least at present to my understanding, as Prime Ministers are largely required to appoint Justices from candidates selected by an independent commission—, and have a mandatory retirement age. The strategy of obstructing and playing the waiting game to win over a single government institution to achieve long term changes in the law is much less likely to succeed in the UK than in the US, especially when there are ostensibly much easier avenues to achieve your goals anyway, and they can be much more easily reversed.
