Ignoring the myriad of issues which would prevent any American occupation of the colonies of Upper Canada and Lower Canada circa 1812-1815, the best scenario the Americans can get in any conceivable universe is a repeat of the 1775 campaign where they march on Quebec. From there its hold on in the face of the inevitable British counter attack. Assuming they whether that storm and negotiate for control of the provinces of Canada you have a positive short term situation and a long term headache.
To whit, the debate now becomes whether to admit both new conquests as states or administer them as territories. Upper Canada you could conceivably get away with administering as a territory (80,000 in 1812), but Quebec (335,000 in 1814) will not submit to being a territory for a long time. In this period the French had an almost irrational fear of American institutions and culture, which will play poorly in both Quebec and Washington. Upper Canada is a wild card in whether it would swing towards being pro-US or anti-US in the future.
The addition of Quebec though, causes an unbalance in the free/slave state problem, and adding Upper Canada makes things worse. Most likely you're going to see an earlier source of friction between North and South. The worst part is that Britain now sees the US as a threat to its remaining colonies, and one which should be shoved in line hard when the chance is given. Most likely meaning that any internal problems in the US will see Britain working to exacerbate them. When the slavery question rears its ugly head (could be as early as the 1840s in this scenario) depending on the intervening years you could see rebellions in the South and North as Southerners yearn to break off for their peculiar institution and the Canadians yearn to break off for Church and Crown, depending on who you ask.
Worst case scenario is this leads to a balkanization of the US sometime in the 1850s.