I don't see why it would have made a direct difference. There was no particular recorded Jacobite sentiment in the then colonies, and the strengthening of such sentiment that might have arisen from this turn would thus have had little to feed on. The cascade of unknown consequences that would arise from having different people at the helm could have led anywhere, of course. One can envisage the exact reverse of the present position, with the British conquering Canada in some war or other that didn't happen in reality, and then some or all of the colonies that in actual history remained loyal, developing towards freely given independence under a shared Crown, finding some reason to rebel while Canada remained under that Crown, even to today! Perhaps resentment at accommodations made to Catholicism in the newly conquered territories could be a factor in both revolt and Canadian loyalty, it is impossible to tell.
A matter that for more direct reasons might have gone differently, and not near so well, is the union between England and Scotland. As we know, this was completed near the end of Queen Anne's life, driven in part by her wish to see her son's position in all realms absolutely secure, in part by a sentimental desire as the last Stuart monarch of the Isles (absent an unlikely Jacobite triumph) to complete the project of the first, her great-grandfather James I and VI, whose desire for a formal rather than personal union was, like most things he attempted, frustrated by the English parliament.
As far as ministers and the legislatures were concerned, for England it was hardly an economic necessity but centuries of the Auld Alliance had not been forgotten, and strategically it was sensible to ensure against Scotland ever again allying with Continental powers against England. For Scotland, so dependent on trade with and by then subsidy from England, it was advantageous to ensure that trade between the countries would forever be free, and their willing consent could itself be traded for very favourable terms.
And so it was done, entirely amicably, and no one now in either country would ever wish for that union to be sundered. It is possible though to envisage a different dynastic situation leading to a different and less happy outcome. Union most likely would have taken place, and perhaps it might even have been earlier, but it is possible to imagine a sulky Scotland being dragooned into it and never really reconciling itself to the union, even centuries later. Dragooned probably not by threats of military force, but by threats or even application of an economic blockade, disastrous to the Scotland of the day.
The difference would be that while no one in Scotland would object to Anne being succeeded by her own son, any more than in England, they might have felt differently about her second cousin, and also felt that, the succession being no longer prescriptive but determined by law, they could make their own law and their own choices thank you very much. England would definitely never accept this, Scotland could never itself pose a military threat to England but it forming Continental alliances could. The course of events from there is unlikely to be as satisfactory as it in reality was.
While pondering this I have taken the time to examine what Protestant alternatives to the Electress Sophia and her son the Elector George Louis there were. In essence, none. He had siblings one of whom might have been chosen, but how likely is it that one of them would accept the Scottish crown in defiance of their mother and elder brother? And though this could not be known then the only one of his siblings who had issue was his sister the Queen in Prussia, who clearly would not be available even had she lived longer than she did (d. 1705). What was known then was that no one apart from George Louis and his sister had an immediate heir. Due to a horrible marital situation the Elector himself had just two children, his son who naturally would be his successor and his daughter Sophia Dorothea, who in 1706 married her first cousin to become Queen in Prussia also! Her husband was an only child, so there are no spares available from that line either apart from his own children, of whom there were a number but clearly too young to be considered even if born before the end of Anne's reign, which they mostly weren't. And the Electress Sophia was literally the most junior descendant of James I and VI apart from her own children and grandchildren. Absent her England would have had to search for a Protestant heir among the descendants of Henry VII's younger daughter Mary, and Scotland turn to the Hamiltons, as representing a daughter of James II of Scots.
That would obviously be absurd, yet the only other possibility would be descendants of the second, bigamous marriage of the Electress Sophia's brother Karl I Ludwig, Elector Palatine. There were Protestants among them, but it is hard to believe that they would have been considered. Even harder though to see Scotland making a Catholic choice. What could be envisioned is that, without specifying whom, Scotland might reserve its right to make a different choice, perhaps refusing to enact parallel legislation to an English Act settling the succession irrevocably upon the Electress and her heirs, and it all flowing from there. Even though it is very unlikely that the two countries would not have become and remained united, it is perhaps not unlikely that a union less freely entered into from the Scottish side would never have been accepted in the way that it was, with agitation for a return to separate rather than shared sovereignty continuing even to this day.