You bring up a very good point: If the US gained any territory from an invasion of Canada, (or even failed to) as a result of British involvement in this German-American War Canada may not have gone the independence route, or it may have been delayed for decades longer than it was.
Then again, Canadians may have become disenchanted with supplying Canadian bodies for the British war machine as the horrors of the battlefield were brought to their doorstep in what could easily be perceived by them as an entirely unnecessary conflict. If this were the case would Canada have sought independence sooner after the ATL German War, and if so and Britain was unwilling, do we have the Canadian War of Independence? Perhaps aided by a southern neighbor equally disenchanted with British interference in the Western Hemisphere?
United North American Federation?
One problem here. At the time Canada was already basically independent. The Act of Confederation in 1867 essentially gave Canada everything it needed to be independent. The Canadian government simply chose to wait due to the knowledge that it had a lot more safety and influence the closer it remained to Britain.
It wasn't until the Alaskan border dispute in 1903 that the Canadians decided they needed more of a voice when the British supported the US government in regards to the Alaskan border. It was from that point on that Canada started to demand more of a say in international affairs, while still relying on the British military. The Canadian government was simply bereft of opportunities to show off their new found independent attitude until WW1.
Anyways in TTL, if Britain and Germany joined sides, Canada would probably remain close to Britain. Canada's favourite sport is complaining about the US, so if Britain became less friendly with the US, Canada would support it. But Canada would definitely be asking Britain to send over more troops, and may even increase its military spending (although this is dangerously close to ASB, considering Canada's past history with the military).
Now if Britain did send more troops to Canada, expect Canada to stay close to Britain. The majority of Canadians were very patriotic, and even the ones who weren't always said they would defend their homes against all comers. So as long as its not Britain declaring war first, most Canadians would be unhappily supportive.
If Britain refused to send over at least a token force, then things would be different. In that case the Canadian leaders and elites would be supportive along with a large part of Ontario and the major cities, as well as English Quebec. But the West, French Quebec, and and a lot of the smaller towns and cities would likely be more cautious. And if they thought Britain was being reckless these groups may start demanding a distancing from Britain. But it would probably take at least one less than successful war to get Canada to part ways with Britain.
If the war was a success even with a high casualty rate, Canadians would see it mostly as a victory for themselves, and the Glorious British Empire. In this case it would ensure that Canadian and British relations remain firm. If they lost badly, well most of Canada would be a US territory, and the remaining bits would be clinging desperately to Britain hoping they aren't gobbled up next, so it wouldn't matter.
But if Canada lost a lot of people and didn't get anything, or lost some territory but only minor areas, then it would cause a big rift between Britain and Canada. Especially if Canada believed that Britain had deserted them to support Germany.
In this case no North American Federation. More like a Canada armed state grudgingly remaining with Britain, praying that the still much stronger US doesn't decide to finish it off.