Not very. Dewey won by a huge margin at the Republican convention, and two of his competitors - Douglas MacArthur and Harold Stassen - were actively serving in the Pacific at the time and couldn't campaign. Robert A. Taft had been considered a front runner, but decided not to run and threw his support for fellow Ohioan John W. Bricker, who ended up being Dewey's Vice Presidential candidate. The Wisconsin primary proved Dewey's overwhelming popularity over the other candidates and it was a foregone conclusion at that point. Dewey ended up with 1,056 convention votes to MacArthur's 1. Even if Taft had become the candidate, his well known isolationism likely would have hurt him in the general election, as it did in 1940 when he lost out to Wendell Willkie for the Republican nomination. In the middle of a winning war I can't see him doing any better.
In the election itself, the popular vote was relatively close with FDR's 53% to Dewey's 45%. But the Electoral Vote map was another story, with FDR getting 81% to Dewey's 18%. Barring some scandals, I can't see FDR getting unseated. Even if the war goes badly, that would likely be put on military officers or intelligence services rather than FDR himself, as the Pearl Harbor investigations showed.