What if the 13 colonies in America made some sort of compromise with Britain, preventing a revolution? In this TL, the crown is advised by people suggesting that guerilla warfare, while not capable of actually winning a rebellion, could be quite costly, and it is better to give limited autonomy to the colonies to make them stop being so rambunctious. I am not sure exactly what the colonies wanted, but suppose Britain had some face saving measure to give them some of what they want, and the colonists decide that it isn't worth revolting of the rest. How would this change history?
A lot will depend on when this takes place. There were very large shifts in mindset on both sides of the Atlantic between 1765 and 1775. The most important changes were:
- The British reaction to the Boston Tea Party in 1773, which a lot of people felt turned a reasonable movement of civil disobedience into wanton criminality. (Oddly, the burning of the Gaspee didn't seem to have this effect.)
- The colonial reaction to the occupation of Boston and then, especially, the Intolerable Acts, which colonials felt turned an argument against unfair taxation into a vital defence against European style absolutism
- The ongoing dispute and pamphlet war of the early 1770s, which created a strong sense of American identity that didn't previously exist
If it's a late compromise by the British, it will only be a matter of delaying the inevitable. A new American identity has been founded that is on constant vigilance against every step from the British being a danger to their liberty. Sooner or later, a misunderstanding is going to cause separation. The best you can hope for (from a united empire perspective) is a union in name only, effectively a personal union under the crown.
However, earlier changes can certainly cause a coherent empire to continue. The British actually did begin compromising after the initial misstep of the Stamp Act, but as long as they stick to only taxation measures, there's never going to be a complete removal of tension. There's a basic problem that the Brits want to get at least some revenue out of the colonists, while the colonists feel that any revenue-raising under the current imperial setup is unfair. Eventually, the broader issues of the mercantilist system under the Navigation Acts, and the constitutional system with no formal role for the Americans in imperial decision, will need to be addressed.
However, a grand bargain solving all issues in a couple of years is implausible. What you need is some action by the British showing there is enough low level responsiveness to American concerns for the colonists to feel they can have some influence through legitimate channels. Then other issues to have gradual improvements over the coming decades. As long as that process plays out without anything outrageous inflammatory from London, I think it's more likely than not to work out.
So back to addressing your actual question of the long term impact from history, well it will be huge, and I obviously can't write an entire treatise here. But the initial major changes in the next few decades are likely to be:
- No American Revolutionary War, where the British Empire is cut down to size. The British will face the next general European war in a far stronger position, as they will be able to deploy their entire naval superiority against multiple European powers, rather than have to spend half of it blockading the American coast.
- The French likely join the War of the Bavarian Succession on Austria's side, as they will not have the ARW as a get-out of their alliance. This likely means the restoration of Silesia to the Habsburg domains. This halts Prussia's astonishing rise in our timeline. It is possible the British join the Prussian side, but they will have limited ability to impact the land war. If they join, they would likely take more Caribbean possesions. If Spain is drawn in, attacks on New Orleans and South American could also happen, which could go either way.
- The Dutch Patriot Revolt likely does not happen. Patriot discourse in the Netherlands was focused around a battle within the elite, mainly through constitutional struggles, before the American Revolution. There was a notable change from the Revolution where ideas emerged that the mass of the general people to overthrow the Stadtholder. This won't happen here, which means no Prussian intervention.
- The French Revolution is very hard to predict, but it's volatility means it will likely unfold differently. The French financial system is unlikely to be much better if the French fought over Bavaria, but the greater popular violence will likely be reduced. One possibility is that the Day of Tiles never happens, which means Province of Dauphin never grants double representation in the provincial estates. That could mean no double representation in the estates general when they are called up, meaning the arguments for the formation of a national assembly never happens. The Third Estate would likely have to join with the lower clergy to get their laws passed, meaning much more religious influence in the end settlement. I also imagine the lower clergy would stop anything obviously treasonous from happening. You'd need to write a new timeline to examine all the ramifications of that!