What affect would the survival of the Stuart dynasty of England have on the American Revolutionary War?
short answer? There would be no American Revolutionary War. At least, not the one from OTL.
That's not to say it's going to be all rainbows and unicorns, but the ARW of OTL was caused by a multitude of factors. Many of those factors were- directly or indirectly- caused by parliament rather than the king. In OTL, from 1714-1760 you had parliament running stuff in the colonies, with a king that either didn't understand the colonies or didn't particularly care. TTL, there's no way that parliament acquires even close to the amount of power it had OTL- I've noted before the irony of the accusations of Stuart absolutist rule being made by an absolutist parliament- because you've got a native born king, who understands the politics of the realm, and who is far less likely to leave things in the hands of the ministers/parliament. When George III tried to claw
back power for the crown in the 1760s it caused problems with parliament who treated him as though
they were the masters and
he the servant. The implementation of the Stamp Act was such a stressful period that it caused George III to have the first attack of his later "madness".
In short, the OTL problems in the 13 colonies would in all likelihood not exist. It would be a different set of problems- and somehow, I doubt it would be ones of representation. OTL that issue was rejected by parliament. To James II and James III, it would be a godsend. It would be a useful way of diluting the British parliament to make it- not so much less effective, but rather less powerful- the same way that it was the conservatives and Légitimists in France who were about expanding the voting franchise, giving women the right to vote and so on. Not necessarily because they agreed with the idea (although the comte de Chambord spoke in favour of both OTL. One speech of his about votes for women was attended by a young Emmeline Pankhurst), but because it was a good way of undercutting the support for the liberals.