Nice ending. Looks like everything else is shuffled as well, including Doctors (another one you could have done was James Bonds - I remember reading that Pierce Brosnan nearly got it when Timothy Dalton did OTL, etc., so there's a lot of potential there).
I like the little touches like Beckett being alt-Portillo and IDS replacing incompetent Howard. Looks like both Rodham and Obama were Republicans all along (what a tweest) which in the former case is quite a well-worn POD, as she was indeed one in her youth.
Also nice touch on the Mossadegh thing having gone differently in the past in contrast with Blair's greatly successful war.
The only bit that doesn't really ring true for me is that the public didn't like Blair because he was a 'grinning geek'. I can't really see this happening - I can easily see the idea of Blair being unpopular then popular as you've done it, but I'd have thought the source of his unpopularity would be a sense of him being out-of-touch, elitist and having more personal background in common with the opposition, in contrast to the down-to-earth Brown. More like John Kerry in 2004 than Al Gore in 2000.
Overall this was a very enjoyable project and one which I think could potentially have more mainstream appeal (among the politically aware UK public) than most of our work on this site.