Let's back up a bit. The original premise was the delay of any war or warlike activities until after 5 November 1940. That implies that the events of September 1939 in Poland went as they did IOTL. All well and good: that led to the so-called sitzkrieg / phony war. Thus, as 1940 progresses, the Germans and French/British/Belgians/etc. do nothing but exchange dirty looks across borders.
At this point it's not so simple for FDR to declare for a third term. He did so IOTL with a real war in progress in Europe: France had fallen, Dunkirk was an accomplished fact, and the threat of another general war looked real. In this instance, that's not the case. FDR could try to make a case for a third term based on a threat of general war, but it's going to be a much tougher sell given that nothing is happening/things have been status quo for months.
Now, counterbalance that with the question of "if not Roosevelt, who?" Forget Garner: he's an old-school red-baiter, and his Texas origins won't sell nationally. Farley is too much a machine politician for many tastes but might be a long shot (IIRC, he'd also have baggage in the form of his faith: anti-Catholic prejudice in '40 was less than it was in '28, but it was still there in the heartland and the south). Though there was decided friction between the man and FDR, I'd guess Paul McNutt would probably be the Dems' nominee assuming Roosevelt cannot make a convincing case for a third term. I'd guess he might take Farley as a running mate to pull in the machines and ethnic votes--and as a bone tossed to FDR.
The GOP is also a tough one to decode. Dewey doesn't have the chops: he's only 38 and has no international experience. Taft is too metallic an isolationist, and that won't play with the party establishment in the northeast. Herbert Hoover offered himself as a candidate in the event of a deadlock; that was a non-starter IOTL and probably would be here also, given fresh memories of his one term. And in the event of no real war in western Europe, I have doubts the Willkie supporters could have carried it off. That pretty much leaves Vandenberg, a diffident candidate IOTL. As cold personally as he was, and with as huge an ego as he had, Dewey might be able to be convinced that his best move would be to serve as VP (with the admonishment that in '48 he'd be only 46 years old, and easily able to run in his own right).
With no shooting in the west, a general perception of less-than-effectiveness for the New Deal (given the 1938 recession), and a still-conservative/isolationist bent in the Midwest, I'd suggest Vandenberg would win a close one over McNutt. That said, it would take Pearl Harbor as an all-out surprise to get the US at war with Japan. If Hitler steers clear of unrestricted submarine warfare and leaves US ships alone after Vandenberg takes office, involvement in Europe might be a tougher sell--but at the same time, Hitler's opponents would be better prepared, so that would counter delayed US involvement to some degree.
As the European theater winds down, I could see Vandenberg unleashing Patton to make an eastward dash, unlike OTL, given Vandenberg's suspicion of the Soviets. I doubt he'd have much hesitancy to use Fat Man and Little Boy on Hiroshima and Nagasaki, either. Then, assuming he's still around, Willkie might well be put in charge of postwar reconstruction, and we'd read of the Willkie Plan instead of the Marshall Plan today.