As I think about it, there is one possibility that could possibly, just possibly lead to a somewhat closer result. In our timeline, something along those lines could have come to pass if the Portugese king had not returned to Portugal at the end of the Napoleonic Wars. Now we can't use our Napoleonic wars to get that result because if the 13 colonies never rebelled the French Revolution either doesn't happen or happens very differently than it does in our timeline.
But lets assume that around the French come out ahead in the Seven Years war. Britain would probably end up in another war with them by the 1780s. This war ends up essentially a draw but bankrupts the French treasury ultimately leading to a French Revolution similar to ours. Now however, the wars of Revolution are not limited to Europe, but exported to America. Britain has to divert resources to protect her American subjects and the French Navy is somewhat more effective than it is in our timeline. The King relocates to America because it is seen as less vulnerable than Britain... Now, in this time line, Britain ultimately becomes the junior partner, but a union seems to be more likely to have been retained.
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Bill