So, it seems like the general consensus is: a definite possibility, but not the most likely.
Now, let us discuss the big elephant in the room, which was mentioned by someone in this thread. What about Quebec? What are the results of no War of 1812 for her, and American settlement in Upper Canada?
My thinking is as followed:
Without the War of 1812, Upper Canada reaches a point where a near-majority of the population are American. Either during a local rebellion (e.g. the Upper Canada Rebellion of 1837) and/or during a large war where the UK is distracted (e.g. the Crimean War), Upper Canada separates from British North America, and joins the United States. Britain probably doesn't care enough to do a large intervention and may decide to cut their loses and demand reparations instead, such as the ceding of northern Maine to British North America and acknowledgement of British control of the Oregon region.
The Maritimes (New Brunswick, Nova Scotia, and Prince Edward Island) become the administrative centre of BNA due to them being predominantly English. Eventually, Canada is formed for the same reasons as OTL (American encroachment and consolidation of debt) and Lower Canada/Quebec becomes significantly more important, to an extent where the elected Prime Ministers of Canada come predominantly from Quebec, unless some form of anti-French apartheid is introduced (an idea that isn't as outlandish as one would think, if the Orange Order takes power in TTL Canada). As there is no chance of a Canadian Atlantic-to-Pacific Railroad, British Columbia remains a British colony rather than joining Canada, but Newfoundland and Labrador may be integrated much sooner.
So, to actually answer your question, here are three scenarios, from most likely to least likely:
1. Quebec becomes the dominant member province of Canada. Quebec separatism is non-existent, and Quebec regionalism (rather than nationalism) is mainstream political opinion. Later on, Quebec-dominated Canada will look for autonomy and patriation of the Constitution much sooner, and may eventually become a republic after peacefully cutting ties to the British Empire. In a reversal of OTL, Canada may face a pro-British, Anglophone separatist crisis.
2. The Maritimes and Quebec strike an uneasy political balance between them. Similar to power-sharing in OTL modern African nations, there is an unspoken agreement that the Head of Government will alternate between a Francophone and an Anglophone. Canada remains part of the British Commonwealth for the economic benefits, but everybody knows the Francophone majority aren't really all that interested in helping the British with anything, including wars.
3. The Orange Order asserts itself early on in Canadian electoral history, and make it harder for Francophones to exercise their democratic rights. The Heads of Government are almost exclusively British and from the Maritimes, and forced assimilation into English culture is implemented in Quebec. This can't last forever, and Quebec may rebel to become an independent republic aligned with the United States. The Maritimes remain part of the Commonwealth and the British Empire regardless if Quebec's separation succeeds or not (and assuming the US doesn't invade).