I'm not sure the outcome can be easily determined, tbh. If such an unlikely and improbable war broke out, it could be a near thing for the US.
Here's my thoughts why.
Noting also that both the UK and America is currently involved in Afghanistan and Iraq, I think things would get interesting.
The UK could presumably withdraw its current troops in those theatres through Europe, while America would have to withdraw through a now hostile Mediterranean, across a semi-hostile Atlantic, being pressured from India and Pakistan through Iran/Afghanistan/Iran (who, while not Commonwealth, would hardly need an excuse to oppose the US).
Most of the currently deployed ground troops would presumably be lost, captured, or at least seriously delayed in returning to the US.
The one distinct advantage the Commonwealth would have would be population and numbers - while only perhaps only just over 100 million between the three most developed nations - England, Australia and Canada - there is over a billion in sub-continental Asia, and millions more in Africa. With much of America's current armed forces now delayed enroute from the Middle East, numbers becomes an issue. Probably the US would have to conscript, even if it could rely on its stalwart allies in Israel and South Korea for some support.
The only way that population advantage could be kept at bay would be through America's initially larger and stronger navy. They would have to maintain two major defensive fleets in both the Atlantic (North and South) and Pacific, and probably a third in the Indian Ocean as well, while its defensive airforce and armed forces would initially be pre-occupied with Canada. Depending on how the US and Canada went with getting troops home from overseas, it could be a longer or shorter conflict. Lets say a couple of months, before larger US numbers eventually overwhelmed the west coast of Canada. Perhaps the UK can send support in time to save the east.
If such a war eventuated, the wealthier Commonwealth parties would also start arming and developing the poorer countries, whose troops and raw mineral resources would be their contribution. There would certainly be the resources to begin massive armament and mobilization of new navies, which while technologically inferior to the US, could cause some serious problems. SAS special forces are widely renowned for their capabilities, and so could presumably cause a few problems in outlying American bases. Lets remember the Carribean mostly falls into Commonwealth territory as well, so there are plenty of small islands to launch strikes from. African nations could begin sending troops from the South Atlantic, even as UK/Canada continued to harrass from the North.
Then there's India, Pakistan, Bangladesh, Australia and New Zealand, either rich in technology, population or resources, who would begin to harrass from the other side.
I think the US would certainly hold out, but be forced into a stalemate. Eventually, on delaration of peace, they would be asked by the international community to return control of the parts of Canada they took over, perhaps retaining command of bases in several Carribean nations that would have earned their wrath.
