It's Arlene Foster, you misogynist.
But seriously, though I haven't the foggiest. I didn't really think too much about who's in charge here and there: there's an implication that a much more right-wing Tory is in power instead of Heath after 1970; and Du Cann and Tebbit together account for the Thatcher government then Labour and the Lib Dems form a coalition in 1992, but that's the only change that I thought much about because it was the only one that seemed relevant to the story.
Otherwise, you may use your imagination. Personally, I don't think it's very interesting.
I admit I took it as a given; the regeneration limit is, to the best of my knowledge, only mentioned three times in the classic run of the series: in "The Deadly Assassin", "Mawdryn Undead" and the TV movie. Perhaps it just isn't established here? Perhaps something happened in the story to resolve it and I didn't think to mention it?
Keep in mind that IOTL, the Doctor's "Eleventh" incarnation was technically the final one due to the War Doctor slotting in between Eight and Nine and then Ten pulling off a non-regeneration and he only received a new cycle for the purposes of resolving this matter.
Maybe later. I have some other stuff on at the moment that will keep me busy.