I mentioned Scotland and NI specifically, so as to avoid an apparent gap for the rest of the UK!
The precise legal framework for the other Commonwealth realms does vary a little.
New Zealand has in force what one might call a “patriated” version of England’s Act of Settlement, so strictly speaking its law operates in tandem with English law, rather than being governed by English law.
In the case of Tuvalu (iIrc), the constitution simply ties the role of Sovereign to that of the United Kingdom; in other words, there is no law of succession. Whoever has the one job, gets the other.
Australia is a more complicated setup because of having seven or more different monarchies. Or not: different constitutional gurus have conflicting views on the point. As I understand it, though, the individual Australian states have their own “patriated” versions of the Act of Settlement, whereas the “national” monarchy is tied to the heirs and successors of Queen Victoria in the United Kingdom. What exactly that means is perhaps uncertain: as a matter of strict law, would another 1936-style abdication require separate federal legislation within Australia?