Note: I mostly made this map to get used to Inkscape so there isn't a hugely detailed backstory behind it.
After the defeats of 1757, Britain doesn't manage to completely reverse the course of the French and Indian War, failing to break into the heartland of New France along the St Lawrence valley, although they do manage to capture the more sparsely populated Louisiana and Illinois areas of New France with the war in North America ending with France ceding Louisiana to Britain while retaining control of the colony of Canada.
Pressure from the American colonists to allow the establishment of new colonies such as Vandalia, Transylvania and Charlotina beyond the Appalachians leads to discontent when the British government refuses to grant the new colonies charters. This, together with other issues such as taxation levels and lack of representation in the British parliament, eventually leads to the southern colonies declaring themselves free of British rule and creation of a new nation - the United States of America - in 1789. The northern colonies though, with the ever present threat of renewed French hostility and lack of direct access to the lands further west, remain loyal to Britain and become a refuge for loyalists from the colonies further south. After years of fighting, the British finally recognise the United States' independence and shortly afterwards form the Dominion of New England, granting a degree of self government and political representation to the loyal colonies.
At least partly inspired by the Americans, France erupts in revolution in 1803 against the absolutist rule of King Louis XVI. A period of ultimately unsuccessful constitutional monarchy fails when the king tries to flee to Canada. Although he is captured and executed, his son Louis Charles is successful and and establishes a rival French government in Quebec City, claiming the throne as King Louis XVII and the French Republic as illegitimate. His successor Louis XVIII, recognising the danger of a further revolution in Canada and the loss of metropolitan France to the republicans proclaims the creation of the Kingdom of Quebec. He also settles a long running territorial dispute with the United States over Illinois county.