To be explained later:
1825 - 1829: Thomas Clayton (F-DE) / John C. Calhoun (DR-SC)
1829 - 1833: Thomas Clayton (NR-DE) / John Sergeant (NR-PA)
1833 - 1837: Andrew Jackson (D-TN) / John C. Calhoun (D-SC)
1837 - 1841: William Henry Harrison (NR-OH) / Levi Lincoln (NR-MA)
As I mentioned in the original post, the most likely ending of a Clayton presidency is Jackson being elected in 1828, but this would reduce the significance of his presidency to essentially being a speedbump on the road to Jacksonianism. A much more interesting prospect is Clayton being reelected, which grants him a better mandate and a more interesting impact (IMO)
The higher (but not Abominable) tariffs enacted under Clayton reduce some of backlash that fueled Jackson IOTL, while ongoing infrastructure programs and expanded foreign trade increase support in the Midwest and the Eastern Coast. Clayton's choice of a running mate, John Sergeant (obscure IOTL but prominent after the Amphyctonic Affair) also helps in Sergeant's home state of Pennsylvania. Ultimately, NY splits as IOTL, while Pennsylvania, Ohio and Louisiana all swing to Clayton.
1828: Thomas Clayton (FR-DE) / John Sergeant (FR-PA) (132 EV) defeats Andrew Jackson (D-TN) / John Calhoun (D-SC) (129 EV)
The election of 1828 is also extremely controversial, as the Federal Republicans had lost the popular vote and only won the electoral vote by the narrowest of margins; there are calls, particularly among Democrats, to abolish the Electoral College but nothing comes of it. The second Clayton term is broadly similar to the first, continuing the expansion of public works in the North and Midwest, public surveys, keeping tariffs high and negotiating agreements with European and Latin American states. The spoils system emerges roughly similar to OTL, Clayton installing loyalists wherever possible. Most notable is the Indian Removal Act or lack thereof, as the federal government sides with the Cherokee, refusing to support deportation and designating the Cherokee as a 'dependant domestic nation' and opening many legal cans of worms. Southern states grow increasing resentful of both tariffs and federal intervention in Georgia, but calls for secession are limited and little comes of it--yet. Overall, the economy does well, raising the popularity of the Federal Republicans.
The Federal Republicans nominate Henry Clay as their presidential candidate, but the choice of vice-presidential candidate is disputed. Ultimately, he selects John Holmes of Maine, hoping for both geographic balance and personal loyalty. The Democrats, meanwhile, raise the same ticket as before and campaign heavily, gathering a coalition of those aggrieved or alienated by the Federal Republicans. Jackson wins.
1832: Andrew Jackson (D-TN) / John Calhoun (D-SC) (193 EV) defeats Henry Clay (FR-KY) / John Holmes (FR-ME) (88EV)
(7 EV (Vermont) for Anti-Masonic Party)
Jackson's presidency is turbulent. The central issues are tariffs and central banking, both of which turn in drawn-out fights in Congress. The Democrats want to reduce tariffs significantly but are fiercely opposed by the N-Rs and some Midwestern Democrats, ultimately ending with a garbled, if reduced tariff that pleases no-one. Disputes over the central bank also turn into a quagmire, with Jackson failing to directly kill it but weakening it to the point of near-irrelevance. This benefits the South economically, but the Northern economy begins to stagnate. Then, in 1836, two years of poor harvest and a currency crisis in Britain trigger a recession (an earlier, milder Panic of '37), causing massive spikes in grain prices in cities across America and general economic downturn, all of which are blamed on Jackson's bank and tariff policies.
Jackson runs again in 1836, now with Van Buren as a running mate in an attempt to secure the north, but is facing increasing unpopularity. The National Republicans nominate the war hero and relative moderate William Henry Harrison, with the much more ideological Daniel Webster as vice-president, hoping to balance the Northeast and Midwest.
1836: William Henry Harrison (NR-OH) / Daniel Webster (NR-MA) (159 EV) defeats Andrew Jackson (D-TN) / Martin van Buren (D-NY) (135 EV)