Born in Arklow, Delany studied in the United States at Villanova University, where he was coached by the well-known track coach Jumbo Elliott. His first achievement of note was a place in the final of the 1500 m at the 1954 European Championships in Berne.

In 1956, he became the seventh runner to join the club of Four minute milers, but nonetheless struggled to make the Irish team for the 1956 Summer Olympics held in Melbourne.

Delany qualified for the Olympic 1500 m final, in which home runner John Landy was the big favourite. Delany kept close to Landy until the final lap, when he started a crushing final sprint, winning the race in a new Olympic Record. Delaney thereby became the first Irishman to win an Olympic title in athletics since Bob Tisdall in 1932, and the last one to date.