> That it's a first-amendment violation doesn't change the fact that atheists aren't allowed - until the laws are successfully challenged and voided (not exactly a trivial task), they're still valid. Just like any other form of censorship.
Unconstitutional laws aren't "still valid" until "successfully challenged and voided". The real question is are these laws enforced? If you file to run for political office in one of these states, and you indicate while doing so that you are an atheist, do the election officials accept your filing and (assuming you meet all the other requirements) put you on the ballot? If a law is clearly unconstitutional, executive officials usually will decline to enforce it–they have an independent legal obligation to obey the constitution, and they want to avoid wasting taxpayers' money fighting lawsuits they know they have no reasonable prospect of winning; but then no court gets to declare it unconstitutional either, since no one has standing to sue–such laws are dead letters that can nonetheless remain on the books for decades, since repealing unenforced unconstitutional laws is rarely a legislative priority, and some legislators will insist on keeping them on the books as a way of making some sort of political statement.
> You're also wrong about the head of state thing. The head of state in the UK has to be Church of England, because the head of state there is also the head of the church.
I'm not wrong about the head of state thing. Section 2 of the Act of Settlement 1701 explicitly bans Roman Catholics from inheriting the throne–"all and every person and persons, who shall or may take or inherit the said Crown, by virtue of the limitation of this present act, and is, are or shall be reconciled to, or shall hold communion with, the See or Church of Rome, or shall profess the popish religion, or shall marry a papist, shall be subject to such incapacities, as in such case or cases are by the said recited act provided, enacted, and established". It doesn't impose such a ban on any other Christian denomination or religion. If Prince Charles converted to Greek Orthodoxy, he'd still be eligible to become King. Section 3 says the King or Queen must "join in communion with the Church of England"–but if, hypothetically, a Greek Orthodox King Charles III was happy to take communion in the Church of England even while remaining a member of the Orthodox Church, he'd meet that requirement. Furthermore, if he refused to do so, he would be breaking the law, but he wouldn't thereby cease to be King–section 3 is toothless because unlike section 2 it doesn't provide any negative consequences for its violation.
> The dominion countries don't care, as they don't have an official state religion.
But the Act of Settlement 1701 is part of the law of Australia, Canada, New Zealand, which is why all three countries had to pass laws in order to change it (in order to remove the ban on the monarch marrying a Roman Catholic under the 2011 Perth Agreement.) It may well be true that politically speaking they "don't care", but it is a part of their law as much as it is part of UK law, so legally enshrined discrimination against Catholics still exists under Australian, Canadian and New Zealand law, and will continue to do so as long as the relevant provisions of the Act of Settlement 1701 remain in force (or unless one or more of them become Republics).
> Note also that the Queen doesn't actually do anything in Australia at all (or Canada or New Zealand).
I agree the Queen is very close to powerless in Australia in practice. But that doesn't change the fact that she is legally Australia's head of state–the Governor-General is not head of state, just the head of state's representative. I said that current Australian law bans Roman Catholics (but not members of any other religions or denominations) from being the head of state of Australia, and nothing you've said disproves that.