For the sake of discussion, let's assume that Hull gets the '40 Dem nomination. His running mate would almost have to be someone from the northeast as a balancing ploy. And that's why I think a Hull-led ticket would have James Farley as the VP nominee. Farley would be entirely acceptable to FDR, and the northeastern bosses (e.g., Hague; Curley). I maintain that any potential loss of votes in the Bible Belt attributable to a Catholic on the ticket would be offset by a southerner leading the ticket. But that may not be the primary consideration.
African-Americans will have just recently come over to the Dems in sizable numbers given the New Deal. What will be the reaction given a presidential nominee from the south (OK, not hard-core south like MS or GA, but still...): will they shift back to the GOP? To hold on to these new Dem voters, Hull will have to make some significant concessions to win their trust. While he's not a race-baiter, and was unquestionably a man of integrity, I have to wonder how far he would be willing to go in this regard.
Now, about the GOP nominee: that Hull was an internationalist is well established. That leads me to think that the battle for the GOP nomination will still largely go as we knew it IOTL; i.e., Willkie is the GOP standard-bearer (I wonder if he might somehow persuade Dewey to be his running mate, since Charles McNary was rather reluctant IOTL? Might be feasible since Willkie was originally from IN (Elwood, specifically) and could have shifted his residency back to Indiana if it looked like Dewey might be interested to avoid the prohibition against both ends of the ticket from the same state.)
So, we come down to election day 1940: FDR will bring all the firepower he can to bear on the election. Meanwhile, the GOP raises (valid, IMO) questions about Hull's age and health: the last president to be that old when taking office was William Henry Harrison, although the analogy isn't really valid. At the same time, the black community may well shift back to the GOP, preferring to err on the side of caution: Hull's stances on race and segregation aren't that well known but his origins in the mid-south are. Long story short, look for the election to be a squeaker of the worst sort, probably not decided until well into Wednesday, 6 November or even Thursday, 7 November. I'd guess that FDR's influence might pull Hull through--barely. It would be the closest election since 1916, and maybe the closest since some time in the 19th century. Hull wins with 290 electoral votes to Willkie's 241.
OK, Cordell Hull is in the White House. FDR has already pledged publicly to support / help the new administration in any way possible as the situations with Japan and Germany deteriorate. As tensions mount, the effects on President Hull become more and more apparent: the photographers and movie camermen who were able to gloss over FDR's physical inabilities can't do that with the deepening signs of stress on Hull's face, or his overall loss of weight. The situation is obviously taking a substantial toll. There's something of a bounce-back, ironically, after Pearl Harbor since then the uncertainty and waiting are done, but that doesn't last long with news from the Philippines. Hull's health is declining, only a year into the presidency. The fall of Corregidor on 6 May 1942 is apparently too much: Hull has a severe heart attack, and dies three days later.
James Farley becomes the 33rd president of the United States.