WI the US won the War of 1812 and annexes Canada?
How would this affect world history? What would the territory that would have become Canada look like today?
Well, for starters the real war goals of the United States were not to annex Canada. Rather, the initial plan was to occupy Canada and use it as a bargaining chip to secure concessions from Great Britain regarding border issues, Native American relations, and the impressment of American sailors. Before anyone responds in a knee-jerk fashion, think about it for a second. The US had just finalized the Louisiana Purchase less than a decade prior and had barely begun to settle it. They had no need for more land, pure and simple.
Furthermore, the War of 1812 was viewed as a massive inconvenience by Great Britain. With Napoleon running rampant in Europe there really wasn't a huge desire to fight America, especially over such trivial issues. Throughout the war, the possibility of a diplomatic solution wasn't that far off. For evidence, look at George Prevost's armistice in 1812 and subsequent attempts at negotiation, clearly he believed that the war could be averted. There are other things I can point to but ultimately one can't escape the fact that the War of 1812 was a needless one that really didn't have to be fought, and had cooler heads prevailed in the US probably wouldn't have.
That being said, I have proposed in the past a way for the US to "win" the War of 1812. Basically, it involves the US being a bit more sane in the strategy department during the early phases of the war. One of the great ironies of the War of 1812 is that for the entirety of the war the US recognized the strategic importance of taking Kingston and severing Upper Canada's supply line to the rest of Canada and by extension the British Empire. Yet due to poor intelligence, indecisive commanders, and spendthrift governments no attack on Kingston was ever mounted in OTL. Our Point of Divergence in this scenario is simple then, in this timeline, while planning for the War of 1812, the US puts an even greater emphasis on taking Kingston than in OTL. Hull's expedition is sent to Detroit, just like OTL and likely suffers a similar fate, however further east the Americans launch their attack. Using converted civilian craft they cross Lake Ontario and seize the vital city before significant forces can be accumulated there thereby cutting Isaac Brock and the rest of Upper Canada off from supply. The taking of Kingston also gives the Americans complete naval superiority on Lake Ontario further complicating matters for the British.
With both Kingston and Lake Ontario firmly in American hands, the British situation in Upper Canada collapses immediately. Brock will undoubtedly try to marshal his troops to try and retake the city. But with no appreciable logistical base and facing a well supplied, fortified American contingent in Kingston he's going to fail. Furthermore, without the constant flow of British supplies to Tecumseh's forces his numbers will dwindle rapidly. He'll probably be coerced by Brock into trying to retake Kingston. For the sake of the scenario let's say the remnants of Tecumseh's forces accompany Brock and attack Kingston resulting in a British defeat and the deaths of both Tecumseh and Brock.
In the aftermath of the failed Battle of Kingston, Upper Canada effectively falls under American control and George Prevost negotiates an armistice (Considering that he did so with the British were winning in OTL, wouldn't he do so if the British were losing in TTL?) As the winter of 1812/1813 draws on both sides are not looking forward to the coming campaign season. Despite having seized Upper Canada and defeated a portion of the British Army, America still faces the bulk of the British Army well entrenched in Quebec. On the other side of the coin, the British are now faced with launching an expensive campaign to oust the Americans from Upper Canada while the war in Europe rages on. Though hawkish politicians on both sides of the Atlantic call for an aggressive prosecution of the war in 1813, in TTL cooler heads prevail, and Prevost's armistice leads to negotiations eventually culminating in an Alternate treaty of Ghent.
America gets what it wants, an end to impressment (basically a non issue by 1813) British promises to stop aiding Natives Americans in the US (also a non-issue with the death of Tecumseh) and promises to resolve other outstanding issues such as borders, fishing rights, etc. in exchange for the return of Upper Canada.
World History probably is only altered in minor ways through the Butterfly effect. The Federalist party in the US will live a bit longer without the Hartford Convention but it won't be able to win in 1816 for sure. The Creek War is much shorter due to no distractions for the Americans. Probably a host of other differences in American history.
As for Canada, especially Upper Canada will be much different than OTL. One thing that immediately comes to mind is the issue of collaboration. In OTL most of the inhabitants of Upper Canada were actually American, lured there pre-war by offers of free land. In OTL there were serious issues surrounding collaboration in the areas that the Americans occupied. In TTL these issues will be writ large across the entire colony. I could see many Upper Canadian settlers, ostracized in Upper Canada post-war due to their collaboration moving south. Also there probably won't be a "militia myth" in TTL due to the previously mentioned factors as well as their "defeat" in the war. Canada remains closer to Britain for longer in this TL I'd imagine. The Family Compact also probably has greater power and influence for longer due to it's presence as a bastion of anti-American resistance during the war.