Everything has happened up to the beginning of WWII as it did in our real history. The difference is that in this scenario, suddenly at the beginning of WWII, everything is exactly the same, except that GB controls part of America and the U.S. does not exist. The change is only at WWII; the time before that is totally unaffected.
So we assume the US east coast is under British control, while the rest of the US is a patchwork of neutral, maybe Spanish states and colonies trading with anyone and helping no one.
Being held as a colony is not as productive as being a free and democratic country. While the east coast is probably quite rich and can muster their own military, it's not the powerhouse it is today with the backing of the entire 51 states. We can safely assume this colony can handle it's own protection with regular troops as well as send some money, elite forces and navy to help Great Britain wherever it's needed.
Great Britain itself has always been an Empire. Another colony will not change that much. It will improve it's strengths a bit because of additional income and manpower, but it's organisation, political actions or general world view would remain unchanged.
So there we are, at the brink of another war. Would it have started? Most likely yes. The Nazis did not start the war with a great master plan in mind. "World War" was a risk they took when expanding eastwards, not their actual goal. Hitler was a gambler and would have taken the risk anyway, even if it would have been greater. After all, there is not much greater risk than the "World War" he was risking already.
In terms of military, Great Britain would be slightly better in what they were already great in: Navy. They'd have more ships and better ships. They already had planes and tanks that were good, if not even better in a plane-by-plane or tank-by-tank comparison than Germany's. With more resources, they might have been even better. However, the tactics lacked. And when it lacks with a colonial Empire with X colonies, there is no reason to assume a colonial empire with X+1 colonies would improve significantly.
Germany won their first campaigns not by being superior in weapon design or even superior in absolute numbers. It won it's early campaigns through superiority in organisation, focus and tactics. "Blitzkrieg" was not about having the coolest tank. It was about ruthless use of combined arms effects, concentration of power, daring leaders and a few elite troops.
Chances are, the campaigns against Poland, Denmark, France, Yugoslavia and Greece, even the Soviet Union would not have changed much. The campaign against Norway relied on seaborne transport. It was clear from the start that the German Navy would have losses and with even more British sea power, more German ships would have been lost. Maybe, Operation "Weserübung" would even have failed. Or never been started. But again, it had been a gamble from the start, and I think it's safe to assume that gamblers will continue gambling even at higher risk.
The battle in the Atlantic would probably have been turned in favor of Great Britain quickly. Being able to protect the convoy routes with their own ships end-to-end using air cover and without the help from a non-combatant that had to hide all their actions as to not be drawn into the war would be an enormous advantage. The convoy battles elsewhere, for example later on the route to Russian ports would have been easier also, with more navy and maybe even Norway free of enemy occupation.
However, while the convoys would be safer, they would also either not transport as much or transport goods at a much higher cost. A lot of the goods that went by convoy either to support Britain or the Soviet Union was lend lease. Without a US, those goods need to be paid for. Assuming the colonies can easily support Britain itself and can also be taxed into oblivion in wartime, that still leaves the goods transported to the Soviet Union.
So in Europe, the course of the war will be slightly altered, with Norway maybe not occupied and Great Britain free to put less resources into convoy raid protection and maybe more into bombing the Reich. With more resources, maybe that tactic would not have been abandoned. However, although not directly involved, with the US and it's economic power out of the equation, not much would have changed. The advantages for Britain would probably be on par with the disadvantages that not having the other US states on their side impose. If you look at who actually suffered the losses of the war and who actually killed the most soldiers and defeated the Wehrmacht, then a change in UK power would not matter much. The Soviet Union did a majority of the fighting, killing and last but not least dying to bring down the Axis.
With Europe only changed in details, the whole conflict in the Pacific would change radically. With the US out of the picture (that would probably include the Philippines, maybe still a Spanish colony or independent then), Japan would rule supreme. Being locked in a land war with China, there would be a huge what if. What if Japan, not threatened by a US embargo, simply decided to conquer China and live happily ever after? Great Britain and the Soviet Union would defeat the Reich more easily. But what if not? In reality, Japanese Ships butchered almost any allied fleet presence but the US navy. With Pearl Harbor a free trading port, the Japanese Navy would be free to chase down any remaining navy in the Pacific and Indian Ocean. Instead of invading the Philippines, they could conquer New Zealand or Australia. They could certainly harass India enough to neutralize it in terms of help for GB in Europe. However, both options are purely speculative. Unlike with Europe, the situation would be so different, we could only guess.
So the bottom line is:
Not much would change in Europe, but the changes in Asia would impact Europe. And the changes in Asia without a US presence are hard to project. It may have went either way. Use whatever fits your story best.