[citation needed] After completing his studies at Marist Brothers College, Parramatta, he was ordained to the priesthood in 1949, on the same day as the future Australian cardinal Edward Cassidy.

In 1953 Clancy earned a Licentiate in Sacred Theology from the Pontifical University of St Thomas Aquinas (Angelicum).

A notable event in the life of his tenure was the beatification of Mother Mary MacKillop, conducted at the Royal Randwick Racecourse on January 15, 1995.

Pope John Paul II referenced Clancy in his homily when he said: "We are celebrating an extraordinary event in the life of the Church in this land: the beatification of Mother Mary MacKillop, the first Australian formally declared to be among the Blessed in heaven.

I rejoice with all of you: with Cardinal Clancy and my Brother Bishops, with the priests, Religious, all of you, lay men and women, families, young people and children, who offer a radiant and authentic sign of the Church's vitality.

Clancy's health began to decline after his retirement and he died on 3 August 2014, aged 90, at the Little Sisters of the Poor nursing home in the Sydney suburb of Randwick.