Guys
The big butterfly, if Britain kept the American colonies is how the rest of Europe responds. Historically the considerable alliance against Britain in the ARW was probably largely triggered by the degree of British dominant of the colonial and economic world, coupled with a period of poor diplomacy on Britain's part. If the conflict is avoided or Britain wins then balance of power politics would probably come into play again later on. The French Revolution could derail such an alliance in the short term but expect a major class sooner or later, especially as Britain continues to industrialise. This could see a British defeat, a draw of exhaustion or substantial gains on the continent itself.
If the American colonies stay with the UK, then another clash would occur between the British and French. I think it would be over the Netherlands, with the Anglo-Prussian alliance supporting the Organists and the Franco-Austrian alliance supporting the Patriots. This war would weaken French finances in the same way that the ARW did, and then the food prices in '89 set off a popular revolution. IMO the French Revolutionary Wars were inevitable.
The French Revolutionary Wars give Britain a chance to pick up all the strategic pieces that it needs for a truly dominant global empire. Even if the French Revolutionary Wars doesn't go into Round 2 with the Napoleonic Wars, the Anglo-American Empire has already taken all the important real estate. India was firmly British since the Treaty of Paris in 1763.
So the 19th century balance of power is going to be tipped even more toward Britain to a really ridiculous degree. The American colonies grow at more or less the same rate as OTL, industrialization might be a little retarded from competition in Britain, but on the other hand there is more opportunity for British investment in North America.
I don't think that the rest of Europe is going to pick up on the whole Anglo-American Empire is a true superpower thing because they will be focused on their own problems. OTL France was navel gazing for most of the 19th century, with frequent regime changes (1800, 1815, 1830, 1848, 1871) and ATL would probably be more of the same. The Germans and Italians are also navel gazing, with wars of national unification being much higher on the list of concerns than Anglo-American global dominance. The Russians really present the only threat to that dominance, and I tend to think the British would heavily and openly back the Ottoman Empire, perhaps even to the degree that Balkan nationalism nevers gets off the ground, along with making sure to block any major Russian advances in Central Asia or China (as per OTL). I think that the British would get the parts of Africa that would be easy to turn into settler colonies, with British colonial policy picking up some of the American need for frontiers thinking. Once the American West starts closing down in the very late 19th, early 20th century, then Africa us ready to open up, with ATL Kenya, Rhodesia, and maybe Tanzania and Namibia.
Anyway, OTL once the Germans unified they presented a challenge to the European balance of power, since they threatened to become the paramount power of Europe, something that Britain needed to watch out for. ATL that doesn't really matter. The British are going to be extremely secure behind what by that time would be the steel wall of the Royal Navy. Even if a German-Russian alliance emerged in order to combat the British, it isn't going to be able to match the industrial might of the Anglo-American Empire. The French could end up with some kind of understanding with the British just to protect them from the Germans. And none of this might matter because at some point during the 19th century the Anglo-Americans probably start pushing for free-trade and have the economic and political heft to force it down Europe's collective throat.
I basically don't think that Europe can really threaten the Anglo-American Empire in a meaningful way. Napoleon's Empire came the closest of any European project to break OTL British Imperial power, and it failed. In OTL what finally broke the UK was WWI, which drained it of manpower, loaded it up with debt, and gave the colonies the chink in British armour they needed to eventually gain independence. With the American colonies manpower and industrial might it won't be possible to beat the British Empire, even if something like WWI happens.