Let's consider two different scenarios.
One is that the 13 colonies don't mind the Townshend Acts with a 1764 POD doing something on public opinion. I think someone in OTL told the British that the colonists hated direct taxes and this was the solution. Well, obviously that someone was a liar or clueless. Anyways, in TTL, it's tolerated with little more than annoyances, except maybe in Boston. Therefore, the American Revolution doesn't happen.
The next one is that the American Revolution does happen, but doesn't really work. POD is 1777, with the French joining in earlier than OTL. However, the British win at Saratoga, inflicting triple the losses they suffer and completing the Albany campaign (making the French look pretty stupid for jumping on the proverbial sinking ship). They also smash the French navy in the English Channel (no, not the invasion of England thread, just naval battles). By 1780, most of the colonies submitted to Britain, giving token payment and oaths of loyalty in exchange for pardons of most of the rebellion except the most egregious rabble rouses and partisans (most Continental Army and even militia were marked differently than civilians and fought like actual soldiers until the time came to run). France has to peace out with some concessions to the British and I can't be bothered to think of the specifics, you can make it up.
In which case does the British crown end up having a stronger influence on the world? The OTL British Empire classic is considered to contain India, that's why the height of the British power is considered post Revolution.
In both of these cases they are left in charge of a large grumbling population across from the Atlantic. In the first case, there is no actual rebellion (or maybe just one in Boston, but obviously Boston vs Britain is a short fight), but the question of North American wartime contribution to the treasury is left unresolved (OTL Townshend Acts specifically were earmarked towards things like colonial judges or stuff not related to war). The latter case, lots of loyalist property was torched by the rebels and a large of money was spent putting down the rebellion. In both cases, they look less impressive as a nation than they did coming out of their win from the Seven Years War.