The key issue, I imagine, is that Mo Udall would probably do old school liberal programs (which, tax rebellion aside, remain and have remained so popular that neither Reagan or Gingrich touched stuff like Social Security) unlike either Carter or Clinton.
Importantly, both Carter & Clinton ran as economic populists (and promptly became moderate to neoliberal in office… remember that Carter was just incompetent, he was quite moderate as President) and I imagine that Mo Udall would to, more or less.
However Udall would almost certainly stay fairly liberal in office and those positions remain popular—more popular than conservative positions…*as long as social issues don't overwhelm.
Essentially a liberal Udall in office—if he doesn't care much about social issues like a good New Deal Democrat—would change the future Democratic Party…more populist on economics, less interested in litmus tests on abortion and quotas and therefore more likely to remain competitive with the Republican Party.
Think a late-forming version of RFK's '68 coalition—the Democratic Party + OTL Reagan Democrats (at least outside the South). Whether or not this remains the case post-Udall is an open question, but President Udall essentially gives the Democratic Party options for the future outside of their OTL coalition of minorities & upper class liberals plus whoever else they can get.
As for the Republicans, without social issues as the wedge they will be unable to embrace supply-side economics unless they want to become as irrelevant as Hoover's Republican Party in the New Deal era.
In all likelihood the Gerald Fords & Bob Doles of the world win out—and the Republican Party stays fairly moderate, if more conservative than in the '50s say.
(This is, of course, all potential. Mo Udall could follow what Carter did IOTL and be a moderate, and care too much about social issues, and so the course of the Democratic & Republican Parties remain roughly the same.)