12 million was the actual number of soldiers the US fielded in World War II. A 9% mobilization rate. Check the numbers and show me where I'm wrong.
http://www.nationalww2museum.org/le...2-history/ww2-by-the-numbers/us-military.html
That war showed the US can mobilize that much people. In a total war with the UK in 1920, the US has a population of 100 million, and using the mobilization rate of World War II, it would be able to mobilize 9 million soldiers, more than the entire population of Canada.
The majority of the army would sit in Canada to wait for the counterattack. If it doesn't come, it would simply annex it and dare the UK to eject them. The rest would be in the continental US, waiting for the invasion that would never come, and if it came, would be swiftly ejected.
And you know what? In that scenario, the US army won't have many casualties. It would be the US navy that would be gutted, but that won't affect what happens in the north.