Kieran Thomas Conry was born on 1 February 1951 in Coventry, England. He was educated at All Souls RC Primary School, Coventry, and Cotton College (junior seminary) in North Staffordshire, before moving to the Venerable English College in Rome to study for the priesthood. He attended the Gregorian University, gaining the qualifications PhB and STB. He was ordained on 19 July 1975 at All Souls Church in Coventry by Archbishop George Patrick Dwyer of Birmingham.

In 1976, Fr Conry returned to Cotton College to teach English Literature and Religious Education. In 1980 he became the private secretary to the Apostolic Delegate (Pro-Nuncio from 1982), Archbishop Bruno Heim, and then his successor, Archbishop Luigi Barbarito. He was appointed Monsignor in 1984. In 1988, Mgr Conry returned to the Birmingham Archdiocese as parish priest in Leek, North Staffordshire. He was appointed Administrator of St Chad's Cathedral in 1990, just before its 150th anniversary the following year. From 1988 to 1993, Mgr Conry was a member of the National Conference of Priests, and its Vice-Chairman from 1992 to 1993. He was also Chairman of the Birmingham City Centre Churches from 1992 to 1993. From 1993 to 2000, he was involved with training counsellors for Catholic Marriage Care. From the beginning of 1994 to 2001, Mgr Conry was Director of the Catholic Media Office in London, the press office of the Bishops' Conference of England and Wales, and also Editor of 'Briefing', the bishops' official journal. In January 2001, he returned to the Birmingham Archdiocese as parish priest of St Austin's, Stafford.