What might have happened if the US had won the War of 1812?

 They would have annexed all of British North America.

The USA would be much more expansionist in this timeline, and Manifest Destiny would pursue. They would claim everything up to the Alaskan border (like what the British did), including the Oregon Territory.

However, the annexation of this much land up north would ruin the balance between Free and Slave states, so the South would be pissed. The USA would need more land for the South, and the Missouri Compromise line would have to be further north. This would happen.

The United States would have pushed to annex Texas immediately. This would kickstart the Mexican-American War. Mexico would still lose, but instead of just taking California, New Mexico, and Texas, the USA would annex the next 1/3 of Mexico, since again, the South would need more land.

Slavery would be more widespread in this world, due to a much larger area and populace being open to it. Most of the American south would be slave states, Sonora, Yucatan, Rio Grande, Kansas, etc.

Once the American Civil War would come along, the South, just like its northern counterpart, would be much larger in this timeline.

The American Civil War would be a much more deadly and devastating conflict, stretching across an entire continent. There would still be border states loyal to the USA though. The North isn’t too disadvantaged, due to their advantage in manpower and being able to pull on Canadian troops. The North still wins, the slaves are still freed, and America rebuilds.

They would still purchase Alaska and might have purchased Greenland as well, as William H. Seward had proposed.

The rest of history is basically the same, but in the Spanish-American War, the USA annexes Cuba as well, due to, well… imperialism. WW1 and WW2 are mostly the same (maybe the USA enters both wars a bit earlier), and the Cold War is the same as in our timeline (just with a larger USA).

Modern-day, there would be many differences between this USA and our USA.

1. There would be some states overlapping the 49th parallel.
2. The Canadian states would be smaller, minus those in the far north.
3. Canada would be more populated, and resource extraction would be much more widespread.
4. The USA would have a much stronger claim in the Arctic.
5. The USA might be a more liberal nation than in our world. Maybe Free or Cheap healthcare would be a possibility.
6. The USA would be the largest nation on Earth.
7. English would be the majority language in Quebec.

And, yeah. That’s what I think would happen.