(Will expand on this post (including the dystopia part) later - gotta get some shut-eye for now)
Well, it is later....
The problem with Canada trying to stand up to the US is that it just can't do so effectively. And even if they choose to side with Evil-UK, it's doubtful that the whole country will be up for a fight - especially the western provinces. Regardless of the sensationalist tales spread by the likes of Henry Ford (who probably just hopped over to Windsor from Detroit) and others, Canadian leftists will be emboldened by the success of their comrades to the south. They'll try and see to it that a Canadian government siding with Evil-UK gets responded to in the same way that MacArthur's junta was dealt with.
As for the European Front:
The Soviets IOTL obviously had a very tough time getting targeted for destruction by the Nazis. But the added power of the UK (which will be obligated to aid Hitler's front in the Anti-Communist Crusade(TM) so as to ensure he doesn't seek revenge against the West for WWI) makes the situation a whole lot worse for them. Most of the European Soviet land will be lost in the initial onslaught, and the casualties resulting from the initial defeat will make Stalin likely to resort to a couple of even more desperate tactics than OTL. In the midst of this, maybe Comrade Mao can lend a hand in the form of Chinese troops (which even in this scenario feels pretty ASB-ish, but with the Motherland's existence at stake, it could happen) sent westward via the Trans-Siberian.
That of course assumes that the Soviets display a preparedness for war similar to OTL.
With the Royal Navy and Kriegsmarine duking it out against the Red Navy, and Imperial Japan blocking the Pacific, the UASR won't be able to deliver most of the Lend-Lease on Steroids help to the Soviets that they want to. Since Vladivostok will likely get besieged by the Japanese, the only real way that American aid can reach the Motherland (until Japan gets taken care of) is via long-distance flights from Alaska to Siberia.
And what of the other European players? How does the Spanish Civil War play out? How does France act here? Do their own homegrown Fascists launch a power grab, and/or does the French Left attempt to make a stand against Western Europe's slide into barbarism? And then there's also Italy to consider...
And Latin America will also be a rather interesting place in this war. I can see Mexico, Nicaragua and Panama swiftly subduing the remaining reactionary holdouts in central America. Meanwhile, South America will be thrown into what'll undoubtedly be its bloodiest conflict as the New World's Anti-Comintern members, led by Brazil, take on the UASR's allies. I'm not sure exactly how this theater would play out, but the UASR and Mexico (the latter moreso given their northern neighbor's focus on Europe and Asia) will surely tip the balance in favor of the Comintern.
In the Asia-Pacific theater:
India will hardly be a "neutral" place in this scenario. The Indian National Congress will likely have its moderates sidelined by its more radical factions in response to London's siding with Berlin.
Considering the difficulty in an American invasion of Western Europe which includes a hostile Great Britain, the focus for the naval effort will be on taking out Japan first and foremost. Assuming that the war starts in 1940, maybe this theater will bear the most resemblance to OTL, depending on how much help Germany and Evil-UK provide to IJN. Also depends on which side Australia and New Zealand choose (somehow I'd expect them to go neutral, assuming that the UASR's Pacific presence is as strong as it was OTL). Japan itself will likely be directly engaged in war w/ the Soviets as part of their Anti-Comintern membership, which even with Moscow's (increased) focus on Europe, probably won't take too much effort to repel.
And why this is just dystopic:
Assuming that the Comintern eeks out a victory over the mega-Axis (still the most likely result), sure, capitalism will be pretty much destroyed along with Fascism. But the body count in the end would probably be similar to that of CalBear's Anglo-American/Nazi War. Great Britain (and France too) will be just as thoroughly destroyed as Germany and Japan were, and the reconstruction process will have totalitarian measures (read: "liquidations" and re-educations) even in areas occupied by the UASR and its Latin American allies. America itself will become more authoritarian due to the pressures of this conflict, with J. Edgar Hoover and his direction of SecPubSafe gaining a reputation similar to that of Dzerzhinski and Andropov.