My round-a-bout path:
Without an Eisenhower candidacy my first question would be whether Dewey runs or not. I see no reason he could not run since his attempt in 1944 could be seen as a "good of the party" run and not a defeat, yet his upset in 1948 must have created serious doubts about his potential as a victor.
My next question is how strongly the Party in that era would seek a "balanced" ticket. From what I have read Nixon was regarded as a moderate to liberal yet had good relations in conservative circles. Thus he might be the "moderate" VP to run with Taft (or the "conservative" to strengthen Dewey, especially if Dewey is seen as needing a heavier handed campaigner as it seems Ike was so he got Nixon).
Assuming Dewey runs and given how close the nomination contest was between Taft and Eisenhower one could see Nixon as the VP for the closely victorious Taft, unless I am missing something, and this might strengthen the call for Nixon to give him some better appeal, i.e. younger, West Coaster, socially "liberal", etc., and assuming the general election puts Taft into office and he dies from illness as he did, Nixon becomes the 35th President of the United States on July 31, 1953 some 193 days after the swearing in on January 20 at 40 years old.
So there is Nixon in '52!