I'll assume somehow the USA wins big in 1812. Maybe intervention in the 1837 Canadian rebellions.
Ontario's Loyalist identity will probably remain in place a lot longer than the Maritimes', which will rejoin its New England kin in local feeling pretty quickly, to the extent these new Maritime states will probably even be called part of New England proper. The Maritimes' Loyalist bit I want to say will be like Rhode Island being a bastion of Quakerism - an interesting bit of history that is decidedly history in this world?
Ontario will join the Upper Midwest/Great Lakes culture, as PA_Dutch said, but will remain slightly more distinct due to the above Loyalism bit. However, the Upper Midwest/Great Lakes areas within OTL America were settled by New England, whilst Ontario was populated primarily by Mid-Atlantic Loyalists (Yankee Loyalists went mainly to the Maritimes) - this gives it a connection to the Ohio Valley/Lower Midwest which was also settled by the Midlanders/Quakers/Mid-Atlantic people. So you may have the Mid-Atlantic Midwest (Ohio River-touching areas), Yankee Midwest (Great Lakes touching areas), and Loyalist Midwest (Ontario), which blends the two with its Loyalist twist. I too also believe the Ontarian Loyalist identity will fade away into a historical thing, the way New York was once New Netherlands, Florida and east Texas were once Spanish, etc.
Alon made an interesting point, I suppose you could have *Ontarian settlers moving west to Winnipeg and beyond; but the Yankees who settled Minnesota, the Dakotas, and Montana may take up *Manitoba westward as well for their pioneering.
The American Pacific Northwest was already heavily settled by Yankees and they'd definitely move north into *British Columbia. We talk about New England-Maritimes connections, Midwest-Ontario ones, but IMO the Pac-NW is the most blurred of these Americo-Canadian blendings. Even more so then the Maritimes being part of 'Greater New England', we'll see the Pacific Northwest as a single ethnocultural region.
Quebec will become its own republic because American nativism will be strong in this period - no Catholics, no non-English-speaking peoples they can't overwhelm a la the Cajuns or Californios. Past being in American economic orbit, it'll be left alone by everyone, which suits it and America just fine.
TL;DR: The Maritimes and British Columbia will definitely meld into being 'Greater New England/Cascadia' seamlessly, while Ontario will blend into the Midwest but keep a bit of its own identity due to its Loyalist past. The Prairie Provinces will take in some Ontarian settlers but be mostly settled by Yankees a la Minnesota and the Dakotas as the Erie Canal and railroads are built.