So let us assume then that in response to the Canadian
Rebellions of 1837 and the
Caroline Affair of the same year, the United States and Great Britain cannot come to any agreement, and so after months of harsh negotiations and recriminations, the new president, Martin van Buren, on the advice of outgoing president Andrew Jackson and overwhelming public pressure, declare war on Britain in say, May of 1838.
The United States army in 1838
was barely 9,000 strong, but let us say that for the sake of argument they've been recruiting and calling for men since February 1837. Let's say it was as strong as it was in 1837 around 12,000 men. They've been recalling as many men as possible, putting men into the ranks and getting regiments of Volunteers ready, so let us say that they've got roughly 20,000 men hauled up for war by June of 1838 with say 10,000 more in the pipeline waiting to get to the field. They've also got auxiliaries and suspected members of a Fifth Column in the form of members of the
patriote movement that had rebelled in 1837 and the Hunters Lodges who OTL attacked Canada in 1838. Because of this, the Americans expect that they will be marching in to Canada as liberators. My guestimate is that they could rely upon maybe 3,000 rebels in Lower Canada (modern Quebec) to take up arms alongside them.
Now against this, what might they be facing? Well the British had also responded to the rebellions in Canada and by 1838 they had put 10,000 regular troops into the country and had armed some 30,000 militia and auxiliaries historically. My guess is you have 10,000 extra regulars moving to Canada post-haste as things deteriorate. Their chief work will be: Defending Montreal, defending Kingston/the St. Lawrence, and protecting Upper Canada (modern Ontario) from invasion. Most of the regulars will be in Lower Canada, so much of the force in Upper Canada will be militia with a smattering of regulars to thicken the ranks.
So what happens next?
The US Army, remembering it's failures from 1812, will probably be much more coordinated in its offensives off the bat and send most of their forces to attack Montreal, picking up rebellious Canadians on the way. That's where the meat of this war will be fought. There's going to be fighting in modern Ontario, on Lake Ontario and across the river, but the war will really be decided south of Montreal.
Now what do I think the results will be? Well, quite honestly it's going to be a British victory. Sending 20,000 troops to Canada, supported by probably 40,000 militiamen while the US is using the same avenue of invasion they have now done twice before, the American army is probably heading for a bruising. The British navy is bigger, they're going to be more prepared for war in Canada and on the seas, and who is going to try and jump to the rescue of the US? Nobody really.
In the end what probably ends up happening is the British take a huge chunk of Maine, settling the boundary dispute in their favor, they snag the arrowhead region and portions of the Northwest, as well as carving a chunk out of northern New York from, probably, Plattsburgh to Sackett's Harbor as they'd thought of doing in the last war, as well as trying to hammer down some other issues.
It's a very humiliating defeat for the US. Canada has a new sense of national identity, the
patriote and republican movements of the 1830s are fairly thoroughly discredited in Canada, and the Loyalists become the leaders of power.
Long term, it will probably be blamed on Jackson and Van Buren, perhaps discrediting the Democratic Party. Britain and the US will have stormy relations, but I think that by the 1850s they'll be able to hammer out a more civil relationship. Canada is a bit bigger and the US a bit smaller.