Because Roosevelt initially meant to step down at the end of his second term, it seems likely that he would have stepped down after the fourth, if the war had ended at the same time in this alternative reality as it did in real life.

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In 1939 he gave his papers to the federal government, as well as part of his family's Hyde Park estate, with the intention of establishing a library there where he would spend his post-presidency. He signed a contract with Collier's magazine to write a regular column starting at the end of his second term, in January 1941. Both Franklin and Eleanor Roosevelt talked about returning to private life.

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It was only the impending war that got Roosevelt thinking he should run again. He had been more attuned to the Nazi threat for longer than any other major US politician, and he wanted to ensure the president remained focused on the effort to thwart fascism. Included in that thought was his hope to preserve the New Deal through the war.

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Incidentally, there's one data point suggesting that this is also the reason he was elected to a third term: Gallup did a poll shortly before the 1940 election, asking voters their preference and a few other questions, one of which was, which candidate would you prefer if you thought there would not be war in Europe, and which candidate would you prefer if you thought there would be war in Europe. Asked to imagine there would be no war, a majority of respondents said they preferred Roosevelt's opponent, Wendell Willkie; but given the probability of a war, a majority of respondents said they preferred Roosevelt. People did think there would be a war, and they did vote for Roosevelt.