It would not have affected the 1960 election that much, except to swing Virginia and MAYBE Tennessee to the Democrats.
It would have had more of an impact on the overall direction of the GOP going forward.
I could see this scenario resulting in a very narrow victory for Richard Nixon with Missouri, Illinois, and New Jersey flipping (perhaps by only a few hundred votes each). I also think that Richard Nixon will still narrowly hold onto Virginia by a Gerald Ford 1976 margin and Tennessee by a Ronald Reagan 1980 margin, as both states were still somewhat anti-Catholic in terms of Presidential preference in 1960.