Depends a fair bit on when and how. For example, FWOAN has Burgoyne winning at Saratoga which effectively ends the rebellion in 1777; only a few hundred British lives have been lost, and the Crown and Parliament can laugh it away as a misunderstanding and a few common criminals. "Cornwallis" in GURPS Alternate Earths 2, by contrast, has the rebellion dragging on til 1783; the British have lost a lot of lives and a lot of money, and had another row with the French in the bargain - they are NOT inclined to be conciliatory.
The bottom line is that Chatham is right - Britain's ability to govern its North American colonies by force is already over ~1776. Too much area and a population already half the size of Britain's, the cost in military expenditures to police it is well beyond what Parliament is actually willing to pay. I expect the colonies rebell again during the Great French Messiness (you'll find people who insist that there would have been no French Revolution without the American Revolution, and they're silly. I ran the numbers, and you know how much longer Louis XVI's government could have lasted if it didn't have those particular debts to sink? 1 year. That's all, just one more year. Would have been a very different French Revolution, probably, but more or less on schedule).
What sort of government(s) would have emerged from that second rebellion is, of course, highly debatable.