So there's two groups of effects that seem to stand out to me. The first is the direct impact on the financials of the ancien regime. It is very expensive to project military force across the Atlantic and that was the straw that broke the camel's back in terms of leading to bankruptcy. Having said that, the British and the French were having wars every 25 years or so regardless, so without the ARW I think another war would have done the same thing, perhaps developing out of the Falklands Crisis or the War of the Bavarian Succession. The other group of effects were the intellectual impacts and how they shaped political thinking among reformers in 18th Century Europe. To me, there are four ideas that matter: the public, nationhood, republicanism and the method of political reform.
Let's start with the concept of the public. Prior to the American Revolution, "the public" was something separate from "the people" in European thought. The public were the intelligent, well-read, well-mannered class of taste and distinction, who followed and discussed the affairs of state. They were usually aristocracy, gentry (in societies where that existed), or the emerging bourgeoise. These people effectively set the parameters of political debate. The people were the rest of the unwashed masses. In the emerging Enlightenment thought, these people were too driven by emotion and basic needs to be part of politics - they needed to be looked after, not listened to. It was the public that should decide what was best for them, because the views of the mob were dangerous. Obviously this distinction was driven by the extensive classism that had been inherited from medieval feudal society and re-emphasized by the nouveau riche, who were keen to show they were separate from their lessers by their education and their refinement. In American society, these distinctions were far less important (at least among white people). An aristocracy had not yet really developed and the bountiful opportunities of the place meant there wasn't a class of wretched poor white people to look down upon. In the US, the conceptions of the people and the public merged. This was advantageous for revolutionaries: those demanding change could both claim to be the public - educated enough to make their cases in respectable opinion - and to be the people - the aggrieved majority that policy should be set for. This approach was immediately adopted by would-be reformed among the Dutch and the French.
Related to this, but a bit different is the concept of nationhood. A nation was previously thought of a community of people that had bound themselves together to form a covenant, either with each other or with someone else (e.g. a King). With regards to political history, this usually meant a set of nobles who vehemently defended their ancient rights from royal power. So, for example, the Hungarian nation was merely the Hungarian nobles (of any language!) that had bound themselves together to constrain the King of Hungary, and participated in the election and institutions of Hungary, at a national or county level. An embryonic idea was that the nation was actually all people in a particular community. The American revolutionaries embraced this idea and made it the primary conception of "nation". Of course, their specific rights had never been formally declared, as was the case with nobles, but they insisted that they had inherent rights that should be respected. You can see why European thinkers embraced this concept.
The third has been more discussed. Previously republics were something that existed at a small level. Preferably a city (there had been plenty of historical communes on this scale), but perhaps a handful of provinces could now do the same, if we look at the Swiss or Dutch examples. But just as Athenian democracy doesn't work beyond a certain scale, neither does a republic. History had shown that you need a powerful monarch to keep things together and ensure the country worked. Just look at what had happened to the Holy Roman Empire or the Polish Lithuanian Commonwealth when the central King was too weak! There was a long history to this in medieval thought: the universe has one Heavenly Father, the Church should have one Pope, Christendom should have one Emperor. This was holy and proper. If you read Dante, there is a reason that Satan has three heads. Of course this conception had been increasingly broken down by facts on the ground for a couple of centuries by this point, but the attitude remained. Power centralized in one person is best. The United States proved otherwise: a republic could work on a continental scale.
Finally, we get to how political reform should be achieved. If you read the philosophes of the mid-18th Century, it is very clear they believe Enlightened governance can only be achieved at the giving of those currently in power. The idea of change coming via demands from below isn't even debated, because it wasn't seen as feasible in the age of centralized, absolutist monarchies. The public intellectuals going to coffee shops were equivalent to Chinese scholar-officials: there role is to persuade the powerful to rule better. Even in places like Holland, with a more distributed power system, the way to enact change was to win debates and votes in provincial and national assemblies, through the existing order. Again, America changed that. These ragtag colonists had demanded change and when they hadn't got their due, they had taken up arms against one of the most powerful Empires in the world - and won! Perhaps all these Kings and Emperors weren't as powerful as previously envisioned. Demands and, if they were not met, political violence, suddenly became a lot more popular. Especially among personality types more embracing of that sort of thing.
So now that we have considered these things, let's think how we could play it out in France. The first thing to point out is that the Dutch actually embraced all of this first. The Patriottentijd was a real change just over the border from France, which had to be put down by the Prussians but shaped a lot of thinking. The anti-Orangist party declared themselves the patriots, represented people of the Dutch nation, and moved from a "through the system" approach to forming militias and taking their rights back. They formed new assemblies, with broader suffrage, based on their natural rights, not historical liberties of the previous assemblies. With no American Revolution, I can't see this happening. Instead the Netherlands is likely to maintain political gridlock, with increasing cynicism and authoritarian creep.
So what about France itself? With no American or Dutch example, they are likely to have less drive to political violence. The first case in OTL in the revolutionary lead-up was the events before and during the Day of Tiles in Grenoble. A meeting of interested people decided to come together and decide to call the provincial assembly - notably, with double representation for the third estate, as was historical in Dauphine. That is unlikely to happen now. Rising bread prices and poor harvests are likely just to cause old fashion rioting, not posing as a provincial assembly.
It's probably likely that the General Estates would still get called a national level. The idea of this institution as the channel for reform had been around for a while, ever since the credibility of the parlements had been destroyed via their (temporary) abolition in the mid-century, even if they were brought back later. It's the natural place to go with the financial crisis hitting. But without the Dauphine example before them, I can't see the concept of double representation of the third estate entering the public consciousness. They will probably be called with single representation and be voting by class. However, that said, the Clergy was highly sympathetic to the Third Estate in our timeline, mainly because it was impoverished lower clergy that got elected as representatives. I suspect what will happen is that the events go off track from the King's demands as in our timeline. The Third Estate makes its demands and needs to work with the First Estate to get them through, probably trading off some favors for Catholic viewpoints, but then still needing the King's signature. It then becomes a negotiation between the King and the Estates over what rights they get for what extra taxation, probably with the nobility getting clobbered. At some point the King gets fed up and closes the thing down. There are some resisters, but nowhere near as many as our timeline. Most go back to their provinces, insisting they have achieved some major goals, and petitions start for another calling of the Estates. A minority of hotheads refuse to leave, but without public backing, the King's men arrest them and they get sent to the Bastille. The fail in the moment, but leave a heroic legacy for later would-be revolutionaries. The regime has enough funding from including the nobility in taxation to go on for now, but it is still horribly inefficient, and Britain's industrial acceleration increasingly leaves France behind...
