This is an interesting question. This could set up a situation where Truman wins in the Electoral College and Dewey wins the popular vote.
The obvious effect would be no Thurmond candidacy. So you start by adding Thurmond's 2.4% of the popular vote and his 39 electoral votes to the Truman totals.
IOTL, Truman's national popular vote percentage margin over Dewey was 4.5%, and his electoral vote margin was 114. With the Electoral College, the key is that Dewey got 189 votes, 77 short of the 266 needed to win. With Thurmond's 39 votes going to Truman, Dewey would need all 77 to close the gap. Adding in the Thurmond votes to Truman's total increases his nationwide popular vote percentage margin to 5.7%, but the increase all comes in states that Dewey has no chance to win.
Truman seems to have received over two thirds of the Black/ African-American vote IOTL. Assume this percentage drops by 25%, which is reasonable, and Truman loses 2% to 3% from his nationwide popular vote margin alone. This wipes out his entire gain from Thurmond and then some. probably bringing it down to about 2%. Loss of white liberal votes would wipe out the margin completely. Its difficult to calculate more exact numbers because some of these votes, particularly the white liberals, would have gone to Henry Wallace, not Dewey. But I think Dewey would have wound up with a very narrow nationwide popular vote lead.
As it happened, the three closest Truman states IOTL were California, Ohio, and Illinois, all of which he carried by less than 1% over Dewey. And the three states combined at the time for 78 electoral votes, one more than Dewey needed to win. If Dewey wins all three states he wins. With no civil rights plank, he almost certainly would have carried California and Illinois, both of which have enough African American and white liberal voters to make the difference. But Truman could have still managed to win Ohio, which has fewer voters in both of these categories.
Dewey's three closest states were Indiana, which he carried by less than 1%, and Maryland and Delaware, which he carried by 1.4%. Thurmond's percentage in Maryland, 1.6%, was bigger than Dewey's margin. So another possibility is California, Illinois, and Ohio moving into Dewey's column, but it not mattering because Truman grabs either Maryland or Indiana.
My suspicion is that the result would have been either a really narrow Dewey win in the nationwide popular vote and the Electoral College, due to his carrying California, Illinois, and Ohio, or a Dewey nationwide popular vote win and Truman winning in the Electoral College by some combination of either holding on to Ohio or losing it but winning either Maryland or Indiana.