He engaged in the diamond business in 1897, was graduated from the Brooklyn Law School of Saint Lawrence University in 1914, was admitted to the bar in 1915 and commenced practice in New York City.
Delaney was elected as a Democrat to the Sixty-fifth Congress, by special election, to fill the vacancy caused by the resignation of Representative John Jay Fitzgerald, and held office from March 5, 1918 to March 4, 1919. He declined to be a candidate for renomination in 1918 and resumed his former business pursuits. He was a delegate to the Democratic State conventions in 1922 and 1924 and was deputy Commissioner of Public Markets of New York City from 1924 to 1931.
He was again elected as a Democrat to the House of Representatives, this time to the Seventy-second Congress, to fill the vacancy caused by the death of Representative-elect Matthew V. O"Malley.
Delaney was reelected to the eight succeeding Congresses, holding office from November 3, 1931 to November 18, 1948. He was reelected in 1948 to the Eighty-first Congress but died on November 18, 1948, in Brooklyn.
Interment was in Holy Cross Cemetery.