Hmm...there's something like this in Orson Scott card's 'The tales of Alvin Maker', where Cromwell survives his fatal illnesses (due to magic) and therefore so does the Republic of GB.
From there goes the plot of Alvin maker. And possibly a cameo by Wolverine >_>.
But at any rate, a (non-magical) longer lifespan of Cromwell could very well see the RoGB outlast. (Maybe have Charles II die in some stupid way - James II is going to be a no-no choice, and so the chance of restoration shoots WAY down).
And yeah, having Great Britain as a Republic would probably make NA easier to keep under control. Not sure whether they'd end up in India, but I suppose it's highly likely.
This (the Cromwellian State) has always seemed an interesting AH to me, and good luck to you =]
P.S. on the 'which sides' front, you'd have to make up your own mind about which countries still support popish kingery, and therefore, which ones piss off the Cromwellians.
If I were you, I'd start from the point of divergence, and come to WW2 when you come to it. It'll come more naturally in terms of self-inspiration then.
Feel free to PM me if you want to gabber about this timeline, it interests me, since i'm thinking of doing something vaguely similar.