Coveney was born in County Cork, Ireland, and was ordained, aged twenty-four, as a priest on 21 February 1959 by Cardinal Luigi Traglia. On 27 July 1985 he was appointed titular Archbishop of Satrianum and Apostolic Pro-Nuncio to Zimbabwe and Apostolic Delegate to Mozambique, and ordained to the episcopate on 15 September 1985. The Principal Consecrator was Cardinal Agostino Casaroli; the Principal Co-Consecrators were Archbishop Gaetano Alibrandi and Bishop Michael Murphy who was the Bishop of Diocese of Cork and Ross.
On 25 January 1990 he was appointed Nuncio to Ethiopia and he became Apostolic Delegate to Djibouti on 26 March 1992 and on 30 September 1995 Nuncio to Eritrea.
He became Nuncio to New Zealand, Samoa, Tonga, and the Marshall Islands on 13 August 1996. His remit was expanded to include Fiji, Kiribati, the Federated States of Micronesia, Vanuatu, and Nauru later that year, and to include the Cook Islands and Palau in 2001. As the longest-serving diplomatic representative to New Zealand, Archbishop Coveney served for a time as Dean of the Diplomatic Corps. He was then appointed Nuncio to Greece in 2005.