I think the whole Jacobite issue dies a quick death. It's only real strength was that it supported the claims of the legitimate male senior line of the Stuarts against the usurpation of the junior lines. Anne of Savoy will never even enter into the discussions after the death of James, as riveting as the personal union of Britain and Piedmont-Savoy would be.
What will be interesting is the fate of James' sister, Louisa Maria Teresa. In 1708, she's an apparently healthy lady of sixteen and she's the Stuart claimant after Anne, as the only other surviving child of James II. Her OTL death in 1712 was smallpox related, so I think it quite possible that that gets butterflied away.
Her biggest problem was that she was a Catholic, but that's not an insurmountable obstacle. The Act of Succession could be amended to allow a regency, or a joint monarchy with all executive power vested in her husband (a la William and Mary), or even that her claim passes directly to her first legitimate child, who will be raised Prtestant.
Given that James could have claimed the throne fairly easily if he had converted, I think its a virtual certainty that Louisa would be rehabilitated and married to a good protestant prince/crown prince/king. No body really wanted the Hanoverians, they were just the only availble choice. Depending on the choices made, (and let's not be boring by picking a younger son) you could get an interesting set of personal union coming from her marriage.
George of Hanover (OTL George II) is already married, so that line won't work, unless Caroline of Brandenberg-Ansbach dies early. So no Hanoverian union. Likewise, Frederick William of Prussia had just married Sophia Dorotea, so that won't work either.
However, Christian VI of Denmark might be worth having a look at. He's a six/seven years younger than Louisa but that can be dealt with. His father, Frederick IV could use some help in the Great Northern War. Britain could use the leverage in the Baltic. Christian's reign in Denmark would be a good few years away, so there would be time to plan for the union. Anne's consort, Prince George, was a son of Frederick III (and a Great, Great-uncle of young Christian IV??) would probably push the union. The Whigs would probably go for it as it would undoubtedly further parliament's position with respect to the monarchy. All going well, the couple could firm up the succession by 1720 or so and the house of Stuart-Oldenburg would be born.
So..eventually we'd we could have Christian I/VI, by Grace of God, King of England, Denmark, Scotland, Norway and Ireland, Duke of Schleswig, Holstien and Lancaster, etc...
If they ever got around to Union talks, it would be very interesting.
Let the butterflies start flapping...