I think the key point here is demographics. Europe's population was growing too fast and Europe could not support the excess population. They had to go somewhere. The Native population was being reduced by European diseases, so there was somewhere for the extra Europeans to go. Of course not all the Natives died, so Europeans used force, and someone had to find an excuse so they could go on feeling good about themselves. Some form of Manifest Destiny would have happened, with or without independence. Perhaps the Natives would have gained some time, but that is all.

By the same logic, the population and economy of America would expand to the point where the colonies would be more important than Britan, and there would be more and more non-British Americans. Either the British government would lose control, or they would have to reach an accomodation with the Americans giving them most of what they wanted, which would amount to almost the same thing as independence.

Beyond that, it is all speculation. I can see two basic paths.

1: The colonies gain independence by force at a later date. They are strong enough to keep European powers from interfering in America. (Spain was a spent force anyway.) Things go much the way they did in real life - perhaps.

2: Britan manages to keep some sort of nominal rule. The colonies do not have a common (British) threat to face, and so the colonies remain politically independent of each other (no United States). Because the colonies are divided, Britan remains very significant as the glue that keeps them together (and hopefully from each others' throats). Also, the crown provides the colonial governments with their basis for legitimacy, which enhances governmental loyalty to Britan.
Economic ties and resulting prosperity also increase loyalty.

Its really too hard to guess about the World Wars. Maybe they never would have happened, because Austria and Germany would be too scared to take on Russia, France, Italy, Britan and America at the same time. Then again, a case can be made that World War I was not started intentionally, and the same mistakes that caused it would have occurred. I think we can assume that it would have gone differently if the “American colonies” had entered the war from the start, unlike the real United States, which entered later. The Austro-German alliance would have been overwhelmed by sheer man power and industrial capacity. Demographics again. If the allies had triumphed earlier in WWI with fewer losses, they might have been less furious, and made a more lenient peace treaty. Perhaps the Germans would then have been less angry too, and the Nazis would never have got very far, and WWII would not have occurred. Who knows?