The U.S. and the U.K. would lose, big time.
Let’s assume Canada sticks with the U.S. (not implausible, I think), just to retain the geographic security of the U.S. Much like WWI, say, there’s a flurry of war declarations. Suddenly it’s the anglo-american axis versus the First and Second Worlds, with Central and South America and Africa declaring neutrality. Also, for simplicity’s sake, assume no nukes.
The U.K. gets the living shit pounded out of it by the combine air forces of Russia, France, and Germany, before the U.S. has a chance to reinforce. U.S. forces in Europe are wiped out fairly quickly (though after a valiant, if doomed, struggle). Numerical superiority simply wins the day here in the early stages, while the U.S. is unable to quickly resupply Britain. The U.K. isn’t occupied, which is difficult to do, but basically unimportant once it’s wiped out militarily; what survives of the British Navy sails for the east coast of the U.S.
Then you get stalemate. The U.S. has to seriously consider continental defence, so its power projection is severely hampered by the need to heavily patrol the Atlantic and Pacific coasts of the U.S. Effectively, this means the U.S. is unable to take the war to Europe, since what’s left over isn’t enough to conquer any but one or two countries, which is immaterial since Russia and China are the economic drivers of the war against the English speaking countries (France, Germany and the others can make significant contributions, but without knocking out Russia or China, the enemy is still fundamentally intact and strong, if only in manpower and bullets terms).
The rest of the world could cobble together an invasion fleet, but it would be risky in the extreme given American naval superiority over its coasts, along with its submarines roaming widely.
With stalemate comes the slow strangulation of the U.S. Trade basically dries up, the U.S. economy goes into the shitter, and unless peace is made quickly, America becomes a hollow shell of herself in a few years and starts suing for peace.
The key point to all this is that, if the rest of the world turned against the U.S., the U.S. would keel over fairly quickly for economic, not military, reasons. The strength of the U.S. is built on trade, and while the U.S. can withstand some instability in the world, it fundamentally requires a large number of trading partners to keep going. Look at how talk of a very limited war gives the economy jitters.