It’s so hard to answer these sorts of alternate history questions, there are such butterfly effects.
If Britain effectively cuts off US Western expansion, it will eliminate the USA effectively or at least what we would become, even as a British holding. Assuming no American Revolution, it seems wise to allow the colonies to expand westward and create a British version of the continental USA. If that doesn’t happen, I suppose that we might end up with Spain and/or France holding more of North America. Even if they go along with western expansion, Britain’s policies may not foster growth the same way the USA’s did.
If Britain takes what is now the USA, it means a vast amount of land, wealth, and productivity is now in the hands of the British Empire. As you say, the power is going to shift from the British Isles to North America unless the British Empire tries to disenfranchise the Americans and we see how that turned out historically.
The British were more anti-slavery sooner than the USA and that means that we may see abolition of slavery in the American colonies before the 1860’s. Though this alternative British empire might end up abolishing slavery later than they did historically because they would have greater investment in it from controlling the American south. I imagine that when slavery is abolished, there won’t be a Civil War, because the slave states (assuming there are such at the time) would be so outmatched by not only the north but by the rest of the British Empire. Avoiding the Civil War would save over half a million lives and who knows how much resources, it would also prevent a lot of bad blood between the north and south and maybe between blacks and whites too.
Assuming we go with my above musings, by the early 1900’s, the British control what would have been the USA, all of the British Isles, Canada, Australia, India, huge portions of Africa, and valuable territory in East Asia. With the greater resources that almost all of North America brings them, they might have been able to win WWI more quickly and decisively, maybe resulting in fewer deaths around, subsequently less harsh treatment of Germany, and no Nazi Germany, Soviet Union, or WWII.
Moving forward to today, this British Empire would have the wealth and productivity of the USA as well as the population of India and the resources of half of Africa. It’s very likely that India and those African colonies would be more developed and stable as well under wiser British (and largely American) direction. This nation would be far and away the most powerful on Earth baring some other strange series of events that might bring them low or elevate some other nation up.