I think that we can generally agree that the only way for Washington to NOT become president is if he dies from some natural causes or the like between the Constitutional Convention and the 1st election. Granted that, the front-line choice comes down to Adams vs. Jefferson. There will be little sectional support for either side this early, and with some back-room dealing, or just blind luck, Jefferson will become President.
The question is: what happens then?
I think I answer that question quite convincingly.
Well, to most people France was a wonderful country: the most enlightened one in Europe until the Terror, and this did not really start reeling until well into Jefferson's administration.
The French executed their King (1792), and thus got all of Europe going to war with them. While Americans may agree with them in spirit, the fact is that the US is not in a place where it can contemplate going to war with every major power in Europe. Plus, with the declaration of the French Republic legally ends the Franco-American alliance, since the alliance was with the
Kingdom not the Republic, of France. The Terror badly scared the American elite, and visions of guillotines in American cities were used by the Federalist press just as effectively as the Democratic-Republicans used the specter of a Hamilton-Adams monarchy.
Jefferson, of all people, knew how important compromise was to the creation of the country, and therefore how important it would be once the government had been formed. He also (to some extent) admired John Adams, who would be the vice-president, and so the administration would run significantly similarly to Washington's at first.
What gives you this idea? Jefferson was almost immediately in the opposition against Washington's (that is Hamilton's) policies.
Considering how important a role Hamilton had played at the Constitutional Convention, and considering the lack of other significant banking ideas, Jefferson would go along with Hamilton, if only because there were no alternatives. Also, remember that the Federalist/Democratic Republican split was in effect due to the personality and idea differences between Jefferson and Hamilton - this difference will in fact be downplayed with Hamilton serving in the cabinet of Jefferson (and he will: because of necessity, Jefferson will try to create a cabinet from the whole country).
Jefferson and Hamilton had two very different visions of what the United States was, and should be. That is why they became the main leaders of the Federalists and Democratic-Republicans. Hamilton's vision of a urban, industrial future, with a strong federal government, was fundamentally opposed by Jefferson's vision of an rural, agarian republic with a weak federal government.
Hamilton's financial policies were designed to establish the federal government as the strong national government that he envisioned. That is why he was so strongly opposed by Jefferson and Madison. Therefore, I do not believe that there is any possibility that Jefferson would pursue anything remotely like Hamilton's financial policies.
They were vehement enemies in Washington's cabinet, and in this ATL they would be, if it were possible, more bitter enemies.
The next leading Federalist in Real-TL was Adams, but as Vice-President in a time of great trials he would neither openly nor covertly support anything against Jefferson.
Adams was a great patriot, and I would say that he, like Washington, was truly a man above factions. Having said that, also like Washington, his ideas did lay in the direction of the more Hamiltonian vision of a strong national government and strong national defense. In OTL he was repelled by the partisan political warfare, and deeply hurt by Jefferson's pet editors' attacks on him. In this TL he will probably retire in 1792, rather than continue to participate in a Jefferson Administration whose policies he is increasingly at odds with. He may go into the Senate, where I think he would become an enemy of the Jefferson's faction, and an (unwilling) supporter of the Hamilton-led opposition.
What will be interesting is the election of 1792. Surely Jefferson will have done a good job, but with increasing unrest in France and a lack of unification as had by Washington, there is no guarantee that Jefferson will be re-elected.
I don't think that he will lose the election. Its a total toss-up for who would become President, but I would count out Adams. He won't have Hamilton's support, and Jefferson will be running again. I don't think that Jefferson would have done anything glaringly anti-Federalist, and so without a big-name candidate who Hamilton believes that he can control, Jefferson wins the '92 election.
In fact, probably he and Adams will switch places, thus starting a precedent of 1-term Presidents. Thus, there can be no 'Virginia Dynasty', and indirectly, New England will hold onto more power - therefore there will be no major push for a Federalist Party, and the country will pass directly from the pre-party system to the era of good feelings; to unravel sometime in the 1810's-20's.
The French Revolution spawned a series of wars known as the "French Revolutionary Wars". Part of these wars included both British and French attacks on American shipping, something that caused deep divisions in the United States. OTL Hamilton's Federalists pushed for war with France in Adam's only term, but Adam's sacrificed his political career in order to prevent war with the United States.
Jefferson's election to the Presidency will not butterfly away the French Revolutionary Wars. So we can expect that the British and French attacks will influence the course of events ATL.
Hamilton felt very strongly about where he wanted the United States to go. Jefferson's policies as President would be in direct opposition to this vision. So I think we can expect a 1790's that would, if anything, be even more viscously partisan.
As part of the great compromise of 1790, a new capital city will be built at Head of Elk on the Chesapeake, so that by 2000 the city (called the District of Washington, duh) will connect the Susquehanna and Delaware Rivers across the states of Maryland and Delaware, and the municipality will be the largest on the globe (

).
The "great compromise" was between Hamilton and Jefferson. It was "great" because they were very committed to its total passage (hamilton) and it total defeat (jefferson).
Look up the politics of the 1790's. Then imagine that the man who most held largely responsible for making the United States a reality (Washington) wasn't participating in it. You'll see that my scenario is probably pretty plausible.