Harding's corruption is found out early?
Harding himself wasn't corrupt; that is, Teapot Dome (using that loosely as an umbrella label for the oil lease scandals) was something rigged and executed by members of his administration (Denby; Fall; Daugherty). Indeed, it's well documented Harding pretty much blew a few gaskets when he found out his Ohio friends had done so: there are multiple accounts of him pinning Daugherty against a wall, gripping his lapels, screaming "you double-crossing bastard!" in his face, and all but performing some free-hand dental work. Further, Harding acknowledged that it was his friends who were keeping him up at night, and who had betrayed his administration. To be sure, Harding was weak, but Alice Roosevelt got it right when she said "Harding wasn't a bad man. He was just a slob."
If Wilson's stroke in 1919 had proven fatal, it's debatable whether Marshall would have won the nomination in 1920, although he would have contended for it. Could McAdoo have gotten it? Sure he could, but could he have been elected? Very questionable, given the mood of the nation. But let's suppose that the GOP nominates someone who isn't as malleable / appealing as Harding--like Philander Knox, who was offered the nomination.
Problem is, Knox was almost as much a Wall Street lawyer as Elihu Root. So...McAdoo plays on the anti-Wall Street sentiments of the south, plains, and western states and hey presto, a southerner gets into the White House. And I mean a southerner who lived the first 30 years of his life in the south.
If you thought blacks didn't have it so good with Wilson in the White House, I suspect you ain't seen nothin' yet with McAdoo. The KKK will recognize they have, if not an outright friend, a sympathizer. I can imagine what McAdoo's reaction would be to the Tulsa race riots in 1921...probably something along the lines of "...they probably did something to provoke it..."
A president aligned that closely with the forces of bigotry will probably cause some seismic shifts, resulting in what look like very odd political alliances to us in the early 21st century. Ethnic voters, not the least of whom were Irish Catholic, would be put off big time, and would likely look to see what they could get from the GOP. Same goes for blacks--at least where they could vote. African-Americans would probably be more wedded than ever to the Party of Lincoln. Northern Democrats would be doing all they could to distance themselves from McAdoo, with varying degrees of success.
What I would anticipate is that McAdoo's cozy relations with the KKK would have been enough to make him a one term president--and to sour the electorate in industrialized / urban states against the Dems for some time to come.
Guessing McAdoo's successor might well have been someone who would be sufficiently outspoken to challenge McAdoo on any and all fronts: Charles Dawes. With Dawes in the White House, one has someone astute enough in banking and finance to recognize that a bubble was forming and steps needed to be taken promptly. Dawes' measures might well have mitigated the bubble bursting in 1929, although I don't know that it would have been avoided altogether. I would guess, though, that the presidency of Herbert Hoover might have been butterflied away--and that if he had run in 1932, FDR's campaign would have been decidedly different, since there wouldn't have been such dire economic condidtions to campaign agains.