James was shipwrecked on his way to Scotland during one of the exclusion crises, and there was extensive loss of life, so that's a potential POD for bumping him off in this period before his accession.
It doesn't really matter who comes after Mary (Or Mary and Anne, if Mary dies early as she did IOTL) - I think the salient point is that the crown's still extensive powers would, without any Glorious Revolution (sic) be very substantially retained. The best bet for British absolutism would be to not have the powers of the crown swept up in the issue of 'Popery', as the Whigs succeeded in effecting IOTL - supporting excessive monarchical power essentially became unpatriotic and almost permanently finished off the Tories as a faction, excepting their revival under Anne. Ironically, a succession of Protestant monarchs are much more likely to manage to effect a permanent move in the direction of any kind of Louis Quatorzian settlement than James ever was.
I'd say it's fifty-fifty on whether Mary or Anne manage to produce an heir. (and remember, they don't have to produce a son - just an heir. A girl would suffice.) If they don't, then I can't see a Hannoverian succession with the Tories in the driving seat. Technically, they could pick practically anyone on the line of succession that they could convince to become Protestant, not just the Savoyards. I suspect that Mary being married to a foreign prince wouldn't help matters here; English/British involvement on the continent against the French would likely be weakened ITTL.
I can't see them falling back on the Stuart bastards. Monmouth is certainly out if he's still kicking about, but he's probably stupid enough to try to win the throne by force, which would only weaken the Whigs further. *Berwick is also out being both Catholic and a bastard, sadly so, as I suspect he would be quite effective.