Well, firstly, we have to guess about what the precises length of the term is since it will determine the nature of Presidential influence. A 4 year term and the President's influence is much weakened, an 8 yr term and it's much expanded. There's a nice symmetry to the 4 year term (with corresponding 2 and 6 yr terms for Senators and Reps), so let's assume they keep that. There's also the question of whether and how it applies to the VP in cases of succession.
Before I move on with the WI, I do want to note the preconditions necessary for it. Theorists in the Early Republic and the Founders at the Philadelphia Convention usually eschewed direct term limits and preferred more elections. For example, most governorships were elected yearly, ensuring accountability to the people. There was one big exception: representatives to the Congress of the US under the Articles of Confederation were restricted to serving only 3 years out of any six (they had yearly terms). This points the way to a third option, that chosen by the Republic of Texas, namely the non-eligbility of the sitting President. This reserves the ability of non-consecutive terms. Nevertheless, I'd venture to say that a POD to create such a limit is likely to require the Founders think more about the nature of executive power (OTL they spent the vast majority of their time on Congress). Lastly, the easiest way for this to happen is that Washington decides not to run in 1792; OTL he was reluctant to run for re-election and with some minor changes, might have opted not to run and set a one term standard (similar to the two term tradition of OTL).
Now to the WI. I'll assume a 4-year term with an explicit constitutional limit and a provision for VP succesion similar to our 22nd Amenment.
First question, who is president in 1792? Almost assuredly, John Adams. The biggest difference is likely to be a more explicit leaning toward the Federalists than under Washinton. Either George Clinton or Thomas Jefferson for VP. Here there are important differences. OTL, TJ wasn't a candidate, since he was retiring from Sec State, but with a different rule he might have run as he did in 1796. Odds are against it, I think, because the circumstances that provoked his political desire (the awakening of the Democratic Republicans because of the Federalists tendency towards centralization) won't be present. The biggest change comes if TJ wins in 1796 because this tinkers with which party was in control and all sorts of other effects follow.
The likely result is that the Presidency loses power to Congress. At first, the House of Representatives probably has something like pre-emenicnce as the most 'democratic' branch. Also, at the outset no one has expereince in the government. As things go on, I expect the Senate begins to overwean the House, particularly if it can assert that the power of adivce and consent gives it powers of intiative in matters of appointment and foreign affiars. The events of the French Revolution and Napoleonic Wars only increase the opportunity for the Senate's ascendancy, since national politices (pro-British vs. pro-French) will naturaly find a home in the Senate.
I'd expect that at some point in the 1800s, this balance will shift to the House of Representatives. It takes some time because in the beginning the House was not very well organized as a coherent body to run the government. OTL, it took until Henry Clay. Once the House does organize, it is much better suited to dominate the Senate: it's directly elected, its members have marginally less power, and it can claim original authority on almost anything it can claim involves spending money.
The President does not go away, lapsed into ceremonial ether like those of Israel or Germany. Rather, he probably serves as the primary upholder of constitutionality. OTL in the 19th century, a Presidential veto was assumed to be not so much an instrument of policy as a the main defender of the Constition against the encroachement of Congress. Hence, Presidential Review will likely be much more important OTL (both through vetos and through pardons). There is also the matter of foreign policy, a field over which the President is usually assumed to have large discretion. Nonetheless, a different reading of Article II and a different history to the Senate might significantly curtail that iniative. The last is the most thorny: the President as C-in-C. Firstly, a more energetic Congress can always attempt to push the President to do one thing or another with the army. Secondly, this likely means the President functions like a very powerful Chief of the General Staff/JCS.
I might also expect that rather than OTL's 12 Amendment, in TTL the vice-presidency would tend to be abolished at some point in time and that succession would fall to the President Pro-tempore of the Senate/*President of the Senate as Acting President pending a new election.
The great tension of US Constitutional history is also likely to be feuds between the House of Representatives and the Senate (rather then between the Presidency and Congress). Here again, the Presidency might gain in power as the mediator of such compromises. However, there might also be ammendments to allow for one house to overrule the other or to create an office that would be head of both Houses and thus a PM-like role for the American Republic. I'd expect that the House will likely increase in the number of representatives (this will increase the Power of the Speaker) and perhaps for earlier direct election of Senators.
An alternative route is that the term starts out at 4 years but is later lengthened (probably to 6) to give the President more time to get things done. Note that my initial route assumes that the Presidency does not become the person expected 'to get things done' in normal circumstances; if an effective Congress does not materialize in the vacuum however, a reform may be necessary. The key are the events between 1792 and 1804ish. This is about the time when you'd run out of 'Founding Father' Presidents and about the time OTL when amendments to the Constitution stopped being passed.