I have toyed with a number of possible Anglo-Continental Unions, such as Denmark from Queen Anne, Sweden or Prussia. The most probable union would be Anglo-Dutch which did happen in 1688, and was all set to continue if William and Mary had had an heir.
While it is interesting to speculate how this might have changed English and Dutch culture, in the big stream of History, IDK if I see it making all that big a difference. The main affect would be to make Britain more of a continental power, but preventing France from dominating the Low Countries was the lynchpin of British foreign policy in OTL anyway.
The UK and Dutch were generally allies after 1688, the only exception being the American Revolution. The UK would have had a slight boost with the power of the Dutch navy with it and not against it, but I doubt it would have turned the tide of the American Revolutionary War. Assuming Britain ruled the Netherlands with a light hand, with constitutional liberalism, I don't see a strong anti-English movement. And in a way it is the House of Orange ruling UK rather than vice versa.
Perhaps the French Revolutionary armies would support a Republican anti-English revolt during its war. And a Napoleonic Kingdom might have been set up as in OTL.
Do you see this as changing history in any important way? Granted it makes Britain more powerful with the 2 largest navies in Europe combined, but UK naval dominance was never really challenged, except briefly in 1778. It adds a few Dutch colonies to the British Empire.
But I don't really see it changing much of the course of European and World history. The French Wars, WW1, WW2, European Integration, proceed basically as they did in OTL. The only main difference is the closer cultural, socioeconomic and linguistic ties. Today Holland has the largest English speaking population percentage of any non-Anglo country. And Dutch and English are the closest relatives, combining West Friesian with a French-Latin influence. And they were even closer in 1688, so its possible they could have merged into a single language.
But as far as geopolitics goes, I don't see it changing the general trends all that much. Perhaps it could have blunted the German invasions in 1914 and 1940 slightly.