Assuming that the method of selecting vice-president remains as it was in the original document (runner-up)
1789 - 1795 : Washington
Things are pretty much per OTL. Though, there's more politicing in advance of the 1794 elections then there was before the 1792 elections in OTL.
1795 - 1801: Adams
The key question here is whether Thomas Jefferson comes out of retirement earlier than he did OTL to run for President and hence become Vice-President. OTL the Jay Treaty was a major cause of his re-politicalization and if it occurs per OTL it probably comes too late. If we stick with Washington's term evolving as it did, then I don't think the groundwork is laid for a substantial Dem-Rep. Movement. Instead, I think Hamilton wants to make sure "his man" is in line and hence Thomas Pinkney is VP.
1801 - 1807: Jefferson
Since the Jay Treaty is still likely to occur, I think TJ is going to start moving pretty quickly. He's a far stronger candidate than Thomas Pinckney, whom is probably the Federalists' candidate (though maybe it's Charles). This probably means that the election is not as close as it was OTL. Most likely Burr doesn't have the chance to be duplicitous and the 12th Amendment is not passed (yet). However, Burr probably manages to find ways to get up to no good and ends up being forced out of office. Nonetheless, Jefferson hasn't had the expereince of being Adams' VP. Adams still probably appoints the Midnight Judges, so John Marshall is probably still around.
1807 - 1813: Madison
I don't think it likely that Monroe or George Clinton find substantial support in the DR caucus. Indeed, the latter won't have been VP for TJ's second term so he may not have any support at all. However, he still has the leg up on the VP since the VP and Pres need to be different states.
1813 - 1819:
Okay, this is where things get hairy. Up until now, it's fairly easy to say that things proceed as OTL, with differences only slight. There will be huge butterflies, however, in the lead up to the War of 1812. If an Emargo Act is passed in 1807 it will be Madison that passes it. The pending election in 1812 might seriously de-rail the chain of events leading to the declaration of war in June because unlike OTL, Madison will not be running. You'd only need PA to flip from the OTL results to shift the momentum to the Federalists. If anything, I'd imagine that the election would be more divisive and there will be a new potential split in the D-Rs between the War Hawks and Madison. Given all the butterflies, I'm wondering if you might not have the election thrown to the House. That may be a bit much, though. Also, it's probably going to increase the odds of a New England secession. Probably though Monroe is elected somehow but the War is avoided.
From here on out I won't speculate in detail. In broad strokes, the tempo of US politics will be different.
Washington (F) 1789-1795
Adams (F) 1795-1801
Jefferson (DR) 1801-1807
Madison (DR) 1807-1813
Monroe (DR) 1813-1819
Andrew Jackson (D) 1819-1825 (earlier presidency leads to stronger Whig Party)
John Quincy Adams (W) 1825-1831 (leads to TTL's 12th Amendment)
Henry Clay (W) 1825-1837
William Henry Harrison (W) 1837 (If he was of a mind to get pneumonia, he probably does again)
Daniel Webster (W) 1837-1843
Martin Van Buren 1843-1849 (D)
John Calhoun 1849-1855 (D) (Slightly later Mex-American War; first President impeached, though not convicted)
James Buchanan 1855-1861 (D)
William Seward 1861-1865 (R) (Yes, he causes the Civil War and gets assassinated)
Abraham Lincoln 1865-73 (R-U) (first VP to ascend to office and run again, leading to amendment limiting to one term or eight years; he gets to lead Reconstruction)
Ulysses Grant 1873-1879 (R-U)
Horatio Seymour 1879-1884 (Lib. Rep/D) (dies)
Samuel Tilden 1884-1885 (Lib. Rep/D)
James G. Blaine 1885-1886 (R) (shot)
James Garfield 1886-1891 (R)
Adlai Stevenson I 1891-1897 (D)
Mark Hanna 1897-1903 (R)
Woodrow Wilson 1903-1909 (D)
Theodore Roosevelt 1909-1915 (P)
Elihu Root 1915-1921 (P)
William Jennings Bryan 1921-1922 (D) (he dies early)
Alton Parker 1922-1927 (D)
Robert LaFollete 1927-1929 (P)
Franklin D. Roosevelt 1929-1933 (P)
Herbert Hoover 1933-1939 (R-P)
Robert A. Taft 1939-1945 (R-P)
Thomas Dewey 1945-1951 (R-P)
Dwight D. Eisenhower 1951-1957 (R-P)
Adlai Stevenson II 1957-1963 (D)
Herbert H. Humphrey 1963-1969 (D-P)
Barry Goldwater 1969-1975 (R)
Ronald Reagan (shot) 1975-1980 (R)
Gerald Ford 1980-1987 (R)
Edward M. Kennedy 1987-1991 (D) (impeached and convicted)
William J. Clinton 1991-1993 (D)
Jack Kemp 1993-1999 (R)
William J. Clinton 1999-2005 (D) (the first to pull a Grover Cleveland)
John S. McCain 2005-??? (R)
The political parties get a bit hairy because I'm thinking TR leads a Progressive Party in 1909, which complicates the political picture. They stay around for a while, leading to several joint-candidacies. Also, past about 1861 I'm just having fun.
