What would have to happen for an upcoming British monarch to not be of House (Mountbatten-) Windsor?

Elizabeth didn’t issue the royal proclamation that confirmed the name as Windsor until 1960, when she was already Queen and Charles and Anne had already been born. People such as George Iwi had already concluded that the royal dynasty was now the House of Mountbatten. This did not prevent the Queen from deciding differently. The name of the royal family is subject to the royal prerogative, and it legally is whatever the monarch says it is, even retroactively - to my knowledge nobody asserts that the name of the dynasty really was Mountbatten between 1952 and 1960.

No, it’s entirely a matter of the monarch’s personal prerogative at any time.

Whether it causes upset to traditionalists or whether anyone else particularly cares is another matter.

True (and as someone who goes by different names in different languages, I should know). Isn’t that also still the case with the Pope? He’s known as Francis in English, but I’m pretty sure his official papal name is Franciscus.

Yes, papal names are still commonly translated between languages. As for monarchs, it differs - Charles III is not translated as Karl into German, but I think several other European languages refer to him by their version of that name.

It was a bit of a culture shock, when on holiday in Spain, to see news stories from their UK reporters about “la reina Isabel”.

At least until Parliament decides otherwise.

True, but I can’t see anyone in government being sufficiently bothered to try. There’d always be something more important to deal with.

That goes for pretty much anything in British law - since Parliament is sovereign, there’s no area it could not legislate in, and no provision of law that it couldn’t overrule. But as long as this hasn’t happened, the name of the royal dynasty is part of the royal prerogative, i.e. something the monarch can rule on without the need for explicit authorisation in any statute. Even more so, it’s part of the monarch’s personal prerogative, i.e. something the monarch can rule on without being advised to that effect by the Prime Minister.

The big problem is if Queen Regnant Charlotte decides to be a Windsor and Parliament passes a bill saying that it is now the House of (insert her married name), then Charlotte would have to give Royal Assent to it which she may not not be willing to do.

Yep. I do think it’s important to remember that Parliament has long employed a one-way ratchet of reducing the choices available to monarch.

And leave this valuable British asset unexploited? Surely, some company would be willing to pay a large sum for naming rights. House of Alphabet, or House of Meta?
/facetious

Technically, he was ‘Georg Ludwig’ (or, by his own spelling, ‘Georg Ludewig’). He dropped the second part of his name after 1714, presumably because ‘Louis’ would have sounded foreign in a way that ‘George’ did not. Another example of how monarchs can call themselves what they want.

In Czech, he was referred to az “Charles” while Prince of Wales, but once he became King, they started referring to him as “Karel III. [třetí]”.

This is certainly the long-term trend. An important counter-example occurred in 2022, when the royal prerogative that allowed the monarch to dissolve Parliament at any time (which had been abolished by the Fixed-term Parliaments Act 2011) was reinstated. Of course this was a prerogative that was subject to prime ministerial advice, so in effect the 2022 statute broadened the powers of the Prime Minister rather than the monarch; but the text of the 2022 statute presented this as a reinstatement of the monarch’s powers:

The powers relating to the dissolution of Parliament and the calling of a new Parliament that were exercisable by virtue of Her Majesty’s prerogative immediately before the commencement of the Fixed-term Parliaments Act 2011 are exercisable again, as if the Fixed-term Parliaments Act 2011 had never been enacted.

Fixed-Term Parliaments Act? Never heard of it. You must be imagining it.

One of those “pretend it never happened” statutes…