I'm not particularly sure; bush Did not openly enter elected politics until the 1950s (he dabbled earlier, becoming active in Planned Parenthood and the Republican party in the 1940s, but didn't make his first run for office until 1950, which he lost). His best bet would be in 1968, when the Republicans had a fairly open field. But, by then, he had been out of public office for five years (he left the Senate in 1963) and would be dead in four years (not that he knew that, of course0
Prescot Bush was a leader, albeit not a charasmatic one, of the Liberal/Moderate Republicans in the 1950s, being closely associated with Eisenhower. He was politically close to Rockefeller, but greatly disliked the man, and opposed him at every turn.
Perhaps Bush's best chance would be if either Eisenhower dropped Nixon in 1956 and chose Prescot to replace him on the ticket (Ike, by that point, was looking for a successor, and had deep reservations about Nixon, although his opinion of him was improving), or if Nixon chose him as his Vice-Presidential candidate in 1960.
In the prior case, Bush serves as Eisenhower's Vice-President for four years and becomes the presumptive Republican candidate in 1960. He narrowly defeats JFK that year, after the two split the Northeast (Bush was a deft debator, althoug he can certainly not match Kennedy's youth. Bush's connections to planned parenthood continue to haunt him with the Catholic vote ... but in a year where he is running against a Catholic presidential candidate, he could hardly win much of that vote in any case)
In the later case, Nixon's choice of Bush actually makes more sense than Henry Cabot Lodge. With the election of 1960 so close in OTL anyway, it is easy to see Nicon pulling off a victory. Then, a few years later, a young man by the name of Oswald travels to Illinois where President Nixon is speaking. Prescot Bush is sword in as President of the United States several hours later.