If the American Revolution and any subsequent wars failed, would the thirteen colonies still receive as much immigration as OTL? Would metropolitan cities like New York still hold the title of most diverse city in the world in 2017?
It's conceivable that, part of a transatlantic community, the Thirteen Colonies might receive more immigration, at least from the United Kingdom. Much depends on how stable British America is, mind.
It's also possible that America could end up with similar amount if not more diversity, the British were (slightly) more favorable to the Native Americans so they could be a higher portion of modern America's population and there could be more immigration from other parts of the empire. Although there would probably be more rules about people from other parts of Europe.
Greater restrictions on land, driving up cost, would likely mean slowed Westward expansion. There would be more British/Irish immigration and less from the rest of Europe. The St Lawrence valley is likely to be the route to the midwest rather than New York/Erie Canal.