Yes, alas it is getting Trumpian here.
And yes Musk is already throwing money at political figures in the UK (off the top of my head, he has publicly funded Stephen Yaxley-Lennon aka Tommy Robinson; there may be others I can't bring to mind just now).
It's basically exactly as you would imagine: you may have been out of the loop, but you're not off track at all. Unfortunately.
My hope is that things will go fast enough in the US that the consequences will have dawned good and hard before we next go to the polls here. (Sorry for you guys though.)
I'm not aware of there being any nationality restrictions on a Prime Minister here (though maybe I'm just ignorant). But Musk has massive power here by owning Twitter and having a great deal of money to throw at anything he wants – just as in the US. I mean, he doesn't have elected political office in the US, either. Just the actual office.
As an aside re nationality, Wikipedia says he has at least one grandparent who was both of British citizenship and born in Britain, so depending on Stuff he might be eligible for British citizenship but then again he might not.