Dewey likely would have been re-elected, absent a MASSIVE recession. Even then, he would have been a wartime President, and we have been reluctant to jettison Presidents in wartime.
Dewey was a moderate Republican who, likely, may have been the first MODERN Civil Rights President. Accordingly, it is quite likely that in 1952 there would have been a ticket on the order of Stevenson/Sparkman that would have swept the South but struggled elsewhere. There would have been a rapproachment and restoration of the Dixiecrats to good standing because they would have been needed. There WOULD have been rumblings by conservative Republicans, but not enough to jettison a winning ticket.
Another thing helping Dewey is the usual reluctance to throw out a party after just one term.