I see Angus Robertson, the current Cabinet Secretary for the Constitution, (SNP) addressed this very point on the news last night.I disagree about Scotland, it wants independence not a republic. That would have to be a new different discussion. And if it retained the Pound Sterling, it may have no choice but to retain Charles III as monarch.
For sure NI, if united with Ireland, may not seek to retain Charles III as the monarch.
He specifically said that independence and King Charles III as monarch of Scotland were entirely separate issues. He also went on to say that the Royals have a long and deep connection to Scotland, and, in his opinion, the Scots would retain the monarch as monarch of Scotland also.