Flannery is the eldest of seven children produced by William K. Flannery and his wife Margaret (née Butler). He is educated at St. Flannan’s College in Ennis, County Clare, completing his secondary education at Dominican College Newbridge in Newbridge, County Kildare.
Flannery joins the Dominican Order in September 1944, leading to studies in theology at St. Mary’s Dominican Priory in Tallaght, Dublin, and then at Blackfriars Priory in Oxford, England. Joining the Dominicans he chooses the name Austin and is ordained a priest in 1950. He continues his studies at the Pontificium Athenaeum Internationale Angelicum in Rome. After his studies he teaches Latin at Newbridge College in Newbridge, County Kildare, and then theology at Glenstal Abbey in Murroe, County Limerick.
Flannery edits the Dominican bi-monthly journal entitled Doctrine and Life from 1958 to 1988, while at St. Saviour’s Priory, Dublin, where he also serves as prior from 1957 to 1960. He also edits the Religious Life Review. He publishes many English language documents on the Second Vatican Council.
Flannery’s campaigning to end apartheid in South Africa leads to involvement with Kader Asmal, and the founding the Irish Anti-Apartheid Movement, of which he serves as chairman and president. In the late 1960s his campaigning on behalf of the Dublin Housing Action Committee, due to its association with republicans and left-wing activists, leads him to being accused of being a communist. He is dismissed in the Dáil by the then Minister for Finance, Charles Haughey, as “a gullible cleric.”