Delaney was born on March 25, 1917, in Providence, Rhode Island, and grew up there. He attended La Salle Academy in Providence, playing tackle on their football team and pitcher on their baseball team. After graduating, Detroit baseball scouts told him, "Cut out football. If you don't, it'll cut you out—your fast ball will disappear as those back shoulder muscles contract."[2] He listened, and then enrolled at College of the Holy Cross.
When Holy Cross football coach Eddie Anderson met Delaney and was told what the baseball scouts said, he replied, "Who says football will spoil your arm? Anyhow, you're so big I doubt anything can happen to you."[2] He ended up playing both sports, with the college's football and baseball teams. As a senior in football, he was a first-team All-America selection by The New York Sun. As a senior in baseball, he finished the season winning six out of the final seven games.[2]
In a game against Temple, Delaney, playing on the defensive line, broke through the offensive line into the backfield so many times that opposing coach Fred H. Swan sent him a Temple jersey to wear after the first half ended.[3]
He was a member of the 1938 College All-Star team that played the Philadelphia Eagles in a preseason exhibition game.[3]