Well to begin with Mo Udall vs. Gerald Ford, I believe Udall has a distinct advantage over Ford, much more than Carter.
Carter was a gaffe machine. He has legendarily bad gaffes from the ‘76 campaign trail that shock me! Udall would at his worst wouldn’t discuss his own lust. It’s my honest opinion that Mo Udall would have filled the void if Carter didn’t run.
Udall, being a left-leaning Rep. from Arizona, would probably seek out a moderate Southernern or Northeasterner for the nomination. Senator Lloyd Bentsen from Texas would be a decent choice, probably the best pick put of those who ran for Pres in ‘76.
Udall/Bentsen would be a pretty strong ticket against, Ford/Dole. Udall’s got a ton of personality and Bentsen doesn’t really look boring facing against Dole that year.
Udall would campaign and govern as a different sort of liberal and probably pull that off better than Carter. Despite the folksy air around Udall, he was a Congressman and would know where to begin with working alongside Congressmen (he did write a book on how to be one!) and there’s no chance in hell that he’d forgo a Chief of Staff.
Overall, I think Udall’s term would be similar to Carter (who ran as a conservative and governed as a liberal in a lot of ways), but he’d just be better prepared for the task of it. Plus, butterflies could happen make Iran and any hostage crisis stemming from it a non-issue, which would give Udall an easy re-election run against Reagan (or whoever but it’d probably be Reagan), assuming he remained heathy enough for the office at least until ‘80.
As far as after Udall, if his health is a big enough issue we could see him resigning over it at some point in his second term. VP Lloyd Bentsen in 1984 would be a fine if dull candidate (assuming a good economy and relative peace and stability), but Sitting President Lloyd Bentsen would be a LOT stronger and would probably easily sweep the election.
He also was good enough to be Dukakis’s VP pick in ‘88 in OTL, so if Udall were to resign in his final year we could get a 9 year long Bentsen Administration (assuming that his Presidency is marked by general social and economic stability).
It’d be interesting to see how the Republicans in ‘92 would recon with the fact that their last elected President was 20 years ago and cheated to win (even if he would’ve probably won anyway). I could see the Conservative Movement in the GOP going in and out of fashion by then (‘84 and ‘88 probably having pretty hardcore conservatives, but ‘92 having a moderate as the cons all fail).