He was ordained a priest in 1965, and, as secretary to Cardinal Basil Hume, was appointed an auxiliary bishop, as titular Bishop of Tala, for the Archdiocese of Westminster, in 1986. In this post he had particular oversight of the diocese"s parishes in Central London. On 3 November 1992 he became the 6th Bishop of Middlesbrough.
There was further controversy in 2005 when, during a radio interview, Crowley expressed the hope that married priests might be allowed in the Church within his lifetime.
His resignation from his post ten years before the customary date, which he said was due to the toll of pressures of leadership in a particularly challenging era for the Church, was accepted by the Holy See in accordance with canon 401 para. 2 of the Code of Canon Law ("because of illness or some other grave reason") in May 2007.
As of 2012 he was based in the parish of Our Lady of Lourdes in Wanstead, London.