There are plenty of other Democratic-Republicans and Antifederalists to take over from Thomas Jefferson. Early leaders might be Patrick Henry, George Clinton, Samuel Adams and eventually James Madison. The two party system is going to develop and turn out very similarly than it did IOTL.
As for Washington's Secretary of State in place of Jefferson, we need to look at other Founding Fathers who served in the diplomatic corps during the war. Benjamin Franklin is already dead. John Jay is a possibility. If a northerner is not acceptible for political balance, then I see a southerner who would later be picked for ambassador, like Thomas Pinckney, James Monroe, or John Rutledge. Or he could pick one of the people he later used to replace Jefferson. Marshall would be a good pick, although I think he's not well known enough in 1792 compared to other candidates.
Depending on who gets picked, there be more initial pro-British policies, but that will be counterbalanced by popular backlash as France is more popular with the average person. Washington wanted harmony, so I suspect he's going to want to balance out views to keep a sense of unity in the country. Besides, Washington was not pro-British; he was pro-American and simply understood that British trade was much more important than French trade, and that British presence in Canada was a more important factor than the non-presence of the French.
I still see Adams as the second president, although he may very well be elected to a second term. Hopefully, he will follow Washington's precedent. The 1804 election becomes decisive. Who will it be? Pickering vs Madison?
To be honest, despite the importance of Jefferson, I'm not sure how much of an impact it will have. The USA had a very deep bench at the time. Napoleon may still very well sell the Louisiana Purchase, but maybe no (or a delayed) Lewis & Clark Expedition? A very different University of Virginia though.