And wouldn't a dual citizen conscripted in Canada be subject to the death penalty for treason in the US?
Henceforth you're subject to Lawsuits in both countries, Arrests in both countries by Any or All Agencies, but it also allow you to have a card under your sleeve, that you can surrender any one of them and stay legally in one anytime.
Also, if the two countries have extradition treaties you're subject to availability by one or both of the countries, if they require you on investigation.
In any of the case above happening sadly you can't request Asylum to the other as you're already their citizen. So, you're putting your self to be able to be extracted from foreign soil with or without permission in such a situation, I imagine.
As during war, there are a lot of things that are unstable, you might as well get away with it, if you choose to hide partial truth, based on the condition you're in. Also, I imagine then any paperwork of opposite country would not be valid in one country, unless they want to. Because war is deception, and your citizenship is stability on paper.
The United States would simply roll tanks and troops across the border, making it to Ottawa in fairly short order. Canada would then surrender, and the Stars and Stripes would be raised across all the provinces and territories. From there, Canada could consider itself officially annexed.