Cecil Underwood would most likely win, although LBJ would probably do better in the Deep South and West and might even carry California. On a side note, I always thought that in a scenario where Richard Nixon is elected in 1960 and assassinated in 1963, Henry Cabot Lodge Jr. would have picked Cecil Underwood as his Vice-President when he'd run for a full term in 1964 due to his youth and the possibility of putting some of the Upper South and border states such as North Carolina, Delaware, Maryland, Kentucky and of course West Virginia into play for the Republicans.
In all honesty, this is my attempt at an image reversal of the real 1960 election. On one side is the young and energetic candidate with the party Senate leader as his running-mate and on the other side is the experienced party head who doesn't appreciate the elites that much.