In 1763, after the British won the Seven Years’ War (known as the French and Indian War here in the States,) there was discussion in Parliament over what territorial concessions to take. The obvious choice was Canada; it was France’s biggest and most valuable chunk of land in the Western hemisphere. And that, we all know, is what Britain took.
However, there was a faction in Parliament that insisted that taking the Caribbean island of Guadeloupe would be better. It’s a small island that would give Britain a leg up on the sugar trade, yes, but there was a larger design in play here. The argument was that the threat of the French in Canada kept the colonists to the south in line. If the French were removed from North America, then the colonies would be more prone to erupt into unrest. The faction that made this argument lost, of course, and twelve years later, on the nineteenth of April in '75…
I believe that if Britain had let the French keep Canada, the American Revolution probably wouldn’t have happened—at least, it wouldn’t have happened when it did. Eventually the American colonists would have clamored for more autonomy, just like the Canadians started to ask for in the 1830s and 40s, and like other British colonies asked for later on. However, in the final accounting, this might have meant a smaller chunk of English-speaking America. On the other hand, during the next Continental war, Britain would most likely have taken sparsely-populated Louisiana from the French, as well as Canada, eventually. And who knows? Britain might eventually have seen good cause for a war for the sparsely-populated section of western North America.
Which leads to another question: would that be the Republic of Mexico to the south, or would it still be Spanish territory? If Britain had managed to hold on to North America, would the Spanish have had less trouble holding on to South America? Spain did actively work to help the American revolutionaries in the 1770s, while Britain actively worked to help the South American revolutionaries in the 1820s. Were these independent states in the Western hemisphere inevitable? Or would their colonial masters eventually have seen the need to give them representation? Or was a change in representation actually inevitable for these colonies? Would a better monarch than King George III have mitigated the conflict to the point that there’d be thirteen or so more North American members of the British Commonwealth today? Would the North American stability have helped Latin American stability?
One constant I see in all this is that the French Revolution was coming, and would have happened whether they still held on to Canada or not. That would likely have meant the rise of Buonaparte, too (but, depending on how the Battle of the Nile turned out, he wasn’t an inevitability, either.)
Counterfactual history is always tricky, especially when you move too far along in time from your starting point. The variables multiply with each year, with each month, requiring you to figure more possibilities than you would have time to write down. I believe that if the British wanted to keep everything from Boston to Savannah in line, they would have done well to keep a threat—real or perceived—in the region. They couldn’t have done much better than the French presence to the north.