Um, being born in British North America still makes him British just not English, Scottish, etc (yes, I know it's more complicated than that

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That said I don't think it's out of the realms of possibilities for those born outside the UK but within British Territory to become PM (after all the Isle of Mann isn't part of the UK though of the British Isles). Simply have another person who immigrated to the "home countries" in their childhood get involved in politics.
Or you could promote a tendency for Colonial Secretaries to have raised in the colonies they are secretary of (tho maybe a tad ASB here) and have them invited by the reigning monarch to be PM.
I do believe (if we're following as close to OTL as possible) that the window of opportunity for politicians born outside the UK is limited to the latter half of the 1800s. Earlier than that and they are just not well known within UK to have a power base, later than that are they just are not UK centric enough.
Hope this helps. But also see 037771 and EdT for that period of British history
