I know there are various Jacobite what-ifs relating to the '15 and '45 uprisings, but I think a peaceful, constitutional transfer of power was far more likely by that time, as without a full-scale invasion actively supported by either France or Spain there simply aren't the men or resources to defeat the Hanoverian forces.
Throughout much of Anne's reign she remained in communication with her Catholic half-brother James, Prince of Wales, known to History as the Old Pretender. As a Roman Catholic who refused to recant his faith, James was barred by Parliament from succeeding Anne upon her death, and his titles had been stripped from him.
I seem to remember reading a book at university some years ago that detailed the nature of the letters between Anne and James, and the author seemed to suggest that had James renounced his Catholicism and pledged allegiance to Anne, he would have been allowed to succeed her. This obviously makes sense given that the later Stuarts' primary obstacle to the throne was their refusal to acquiesce to this demand. I can't remember the book off-hand, but I think it was something like The Old Chevalier, and the author's name escapes me. It was a very interesting read, however.
Let us assume that, putting pragmatism before conscience, James finally agrees to convert to Anglicanism and renounce the support of his patron, Louis XIV. Anne, childless and dying, instructs Parliament to allow her half-brother to succeed her, thereby ensuring the peaceful continuation of the House of Stuart by way of the Act of Succession.
Surely it must be considered plausible that, had James agreed to Parliament's demands, he would have been allowed to succeed Anne. It is true to say that after 1689 Parliament were more or less in charge of the show, but would a pliant James III as a constitutional monarch not be a better pick than a German elector nobody had heard of and whose claim - in contrast to James' - was weak?
So, on 1 August 1714 Queen Anne dies and Parliament proclaims James III as King of Great Britain -- James, at court in St. Germains, prepares for his voyage to England, accompanied by his half-brother the duke of Berwick and a small detachment of household troops provided for by King Louis as a parting gift.
How would things unfold?