How would the survival of Queen Anne's son, William, Duke of Gloucester (and his production of issue, male and legitimate) affect the fortunes of the house of Stuart? I've read that the Jacobite risings were caused in part because James III was a nearer relation to the last king (Anne) than George I was and therefore was the rightful monarch.
The line of descent by strict male-preference cognatic primogeniture was:
James II
- his son James "III"
- his elder daughter Mary II
- his younger daughter Anne
-- Anne's son William of Gloucester
then
William III, only child of James II's older sister Mary
then
descendants of James II's younger sister, Henrietta, all Catholics
then
descendants of the older children of James II's paternal aunt, Ellzabeth, all Catholics
then
George I of Hanover, James "III"'s second cousin through Elizabeth's youngest child, who was Protestant
The "Glorious Revolution" deposed James II in favor of Mary II and his nephew William III, her husband. James and his son were excluded as Catholics. With that exclusion, Mary and William, and then Anne were the next in line. With their demise sans progeny, and the exclusion of the Catholic heirs, George was next.
The Jacobite rebellions were based on refusal to accept the exclusion, as a violation of the "true" line of inheritance.
But, if William succeeds as William IV, King of England, France, Scotland & Ireland (assuming that the Act of Settlement and Act of Union do not get passed, since they were passed after his OTL death), then by the law of proximity of blood, he is then a nearer relation than James - son versus brother.
The Jacobites did not argue that James should inherit instead of William; they argued that William III and Mary II, and Anne were all usurpers. James was Anne's brother, and should have been King before her.
The Jacobites actually dated back to the Glorious Revolution and the rebellion led by "Bonnie Dundee". Their rituals including squeezing an orange (in despite of William III of Orange) and toasting "the little gentleman in black velvet" (the mole whose burrow his horse stumbled on, leading to his death).
The reason for the rebellion being in 1715 was that Anne had been more acceptable than George - being British and of the previous royal line. Her acccession had been explicitly accepted. George was a distant cousin and a foreigner, so it seemed possible that there was enough opposition to support a successful rebellion.
Any thoughts on how the Jacobite movement might progress under a half-Stuart king rather than a complete foreigner? Would the Jacobite movement even develop?
The succession from Anne to William would be more acceptable than Anne to George. IMO there wouldn't be enough political energy for a rebellion. The Jacobite movement already existed, but would probably dwindle gradually, assuming the Oldenburg line continued.