Assuming the ARW was not successful and the British were able to hold onto the colonies how do you think the influence and impact of the British empire over the ensuing century plus would have gone?
Would they have become an even more powerful empire than OTL or would their focus and demands change and perhaps not put as much of their efforts into places like India or Africa and elsewhere?
The problem with your what if is that it has no basis.
One way or another, the american colonies will severe links with the London government. It may take 20 or 40 mord years but it will happen.
Not because of so-called american exceptionalism but because of something close to physical laws.
One government can't keep control of a massive population that lives 6000 kilometers away from it, separated by an ocean or more than an ocean.
The pressure for a level of decentralisation that is so juge that it can be called nothing else than "independance" is so strong that it is unavoidable.
That's why Canada, Australia, ... etc, became dominions.
And the other point is that Britain was all but a democracy. It was ruled by a small oligarchy of nobles and merchants that prefered Britain losing territories than themselves losing power. They did not want Britain being ruled by colonials.
Just consider what choice the Penn family did when the american revolution began.