TL;DR: other countries can invalidate court decisions by changing the law more easily.
Many countries use a civil law system rather than the United States common law system. In a civil law system, the clear text of the statute overrides precedent. So if a statute says that it applies under this circumstance, then it is assumed to do so.
In a civil law system, if the legislature doesn't like a court decision, it can change the underlying law. Because courts are more strictly bound to enforce only the statutes that actually exist. In a civil law country, all the judges are expected to be textualists (a position associated with the right-wing in the US).
The United Kingdom (like the US) is a common law jurisdiction. Precedent can override the statute. But in the UK, they have a principle of legislative supremacy. In short, in the UK, the parliament may change any law by a simple majority vote. As a result, if they don't like a court decision, they too can simply change the underlying law.
The US has a constitution that is interpreted under common law. This means that the courts may invalidate laws, overruling the legislature. The only immediate fix for that is a constitutional amendment, which requires the approval of three quarters of the state legislatures. Further, the normal amendment method requires two thirds of both chambers of Congress.
In the longer term, the makeup of the Supreme Court could be changed. And that's what's happening. After the Court's overreach with Roe v. Wade, there arose a strong opposition that specifically wanted to override that decision either by amendment or by picking the right Supreme Court justices.
If the Supreme Court had left that decision to the states (where it had been), then abortion would have stayed a legislative fight. The pro-life movement would be less politically powerful and most people would not particularly care who was on the Supreme Court. But abortion was an issue that people could understand. That case built a political opposition to Supreme Court jurisprudence.
Don't believe me? Here's liberal constitutional scholar Cass Sunstein saying the same thing:
I think that some of the Warren Court's decisions were a little lawless and jumped too far too fast. In so many areas the court's ideals didn't have clear constitutional foundations. The Griswold v. Connecticut case, which created the general right to privacy, was that kind of ridiculously weak opinion. The court didn't identify a clear constitutional basis for saying that the ban on contraceptives within marriage was impermissible. The court referred to "penumbras" and "emanations"[in the language of the ruling] from the Bill of Rights. But the Bill of Rights doesn't have "penumbras" and "emanations"; it just has a catalog of rights. It would have been better to say that the ban was never enforced and it was a recipe for arbitrary and unpredictable action by the police in a way that does violence to the rule of law.
Roe v. Wade itself was probably a horrible moment for liberal politics and almost certainly created the Moral Majority. Roe simultaneously demobilized the pro-choice movement in politics and fired up the pro-life movement everywhere. There probably would've been an Equal Rights Amendment without [Roe v. Wade], less agitation with the process, and stronger legal commitments to sex equality in general. It's absolutely true that if the court goes in the teeth of the public, it can hurt the cause that you're trying to promote.
I think the Court should've said, in the Texas and Georgia cases [pertaining to Roe v. Wade], that these laws are so draconian in their reach that they're unconstitutional. The Texas law didn't allow abortion in cases of rape. So the court could've said very narrowly that we're not going to say anything general about what the Constitution says with respect to abortion -- but women who have been raped have a right to have an abortion. The Georgia law had procedural hurdles for women seeking abortions that seemed to intrude on women's interests and went well beyond what was necessary to make sure that the decision was reasonable and well-considered. The court could've said simply that the Texas law didn't have an exception in cases of rape, and the Georgia law went far beyond what is reasonable and necessary to protect fetal life. And that way there would've been a continuing dialogue between the states and the Supreme Court on the abortion issue.
Well, we don't want to fall in the trap of reading the Constitution to do whatever is good. This is the activist fallacy, on both the left and the right, which says that if something is very good, then the Constitution requires it. Even if the pro-choice people are correct, we have a Constitution that we're reading here. It's not true that the text and history of the Constitution, at the time, clearly supported the broad right to choose abortion. I am not saying that Roe v. Wade should be overruled. I don't think it should. It's been the law now for a long time. But I am saying that as a matter of pure self-interest, decisions like Roe often backfire.
In all, it's a combination of factors. The US Supreme Court is effectively more powerful relative to the legislature than the top courts of other countries where the courts cannot invalidate legislation. Further, the Supreme Court made a blunder by creating a right that had not previously existed that people opposed. Because of the same reasons that make the US Supreme Court more powerful, the only practical recourse that people had was to change the makeup of the Supreme Court. This polarized and politicized the Court in a way that previous overreaches (e.g. Wickard v. Filburn) had not.