The election IOTL was so close that I just assumed when I first read this that a Stevenson-Nixon race would have resulted in a victory for Nixon.
However, checking the Wikipedia page on the election, I'm not so sure. I only see three Kennedy states that I'm confident that Stevenson would not have been able to carry, New Jersey (0.8% margin IOTL), Nevada (2.3% margin), and New Mexico (0.7% margin). These three states combined for 23 electoral votes, not enough to push Nixon over the edge While the closest Kennedy states were Illinois, Missouri, and Hawaii, I don't see Kennedy as having an advantage over Stevenson as a candidate in them. These
The margins IOTL in New York were 5.3% and in Pennsylvania with 2.3%. Stevenson would have struggled to carry New York as opposed to Kennedy, and this would have been enough to elect Nixon. Pennsylvania is different, its always close and it doesn't have a higher percentage of Catholics than the national average.
With Kennedy as a running mate instead of LBJ, Stevenson might carry all the northeastern Kennedy states but New Jersey, but could well lose Texas. With LBJ as a running mate, he probably loses enough northeastern states to be denied the election.
The closest Nixon states were California, Alaska, and Washington. Stevenson does less well than Kennedy in California, and Alaska is a wash. I had first thought he might do a little better in Washington, but then his dovishness would have not gone down well there. Stevenson does pick up the Byrd electoral votes.
It looks like either a Nixon victory, or an electoral reversal, with Stevenson losing the popular votes but winning enough of the northeastern states by very narrow margins to prevail in the Electoral College.