Conway was born on 22 December 1957. He was educated at Archbishop Tenison's Grammar School, a state grammar school in Lambeth, London. He studied modern history at Keble College, Oxford, graduating with a Bachelor of Arts (BA) degree; as per tradition, his BA was promoted to a Master of Arts (MA Oxon).
Conway remained at Keble College to undertake teacher training, and he completed a Postgraduate Certificate in Education (PGCE) in 1981. He then became a teacher, working as an assistant master at Glenalmond College, an independent school in Perth, Scotland, between 1981 and 1983.
In 1983, Conway started at Westcott House, Cambridge, a Anglican theological college in the Liberal Catholic tradition, to train for ordained ministry. During this time he also matriculated, studying theology, at Selwyn College, Cambridge, and he graduated with a further Bachelor of Arts (BA) degree in 1985. After a further year of training, he left theological college in 1986 to be ordained in the Church of England.
Conway was made a deacon at Petertide 1986 (29 June) by Michael Ball, Bishop of Jarrow, and ordained a priest the following Petertide (28June 1987) by David Jenkins, Bishop of Durham — both times at Durham Cathedral. From 1986 to 1989, he served his curacy at St Mary's Church, Heworth in the Diocese of Durham. He was an assistant curate at Church of St Michael and All Angels, Bishopwearmouth, Sunderland, between 1989 and 1990, and then an honorary curate of St Margaret's Church, Durham, between 1990 and 1994. From 1989 to 1994, he was also Director of Ordinands for the Diocese of Durham. He then moved to St Mary's Church, Cockerton, Darlington, where he was Priest-in-Charge from 1994 to 1996 and Vicar from 1996 to 1998.