If you're comparing the politics of abortion (or similar issues) in Britain and the United States then it's an error to assume that the situation in America is the norm and that it is the state of play in Britain that must be explained. On the contrary: what happened here is what happened in most other countries, which is that a pragmatic social compromise was eventually reached on what will always be a difficult issue, one that did not (and does not) exactly please anyone, but which largely takes the heat out of the matter and turns it into a matter of technicalities and regulation. On this particular issue the compromise reached in Britain is actually a little different to most other European countries (essentially abortion is legal for a longer period into the pregnancy, but there is no on-demand access to it) but the general framework is recognisable across the board. So the real question is why this did not happen in America and (perhaps) what this curious failure says about the American political system and its functionality.
A telling thing about the British situation and the extent to which it's been depoliticised is that one of the remaining flashpoints is the requirement under the Abortion Act for two doctors to sign off on a procedure, and specifically that the arguments against it are based as much on
practicality (i.e. doctors seeing it as a waste of their time) as on liberal/human rights grounds.