[1][2] Conway remained at Keble College to undertake teacher training, and he completed a Postgraduate Certificate in Education (PGCE) in 1981.
[1] He then became a teacher,[3] working as an assistant master at Glenalmond College, a private school in Perth, Scotland, between 1981 and 1983.
[1] In 1983, Conway started at Westcott House, Cambridge, an Anglican theological college in the Liberal Catholic tradition, to train for ordained ministry.
[2] 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.
[10] Following the retirement of David Stancliffe as Bishop of Salisbury in July 2010, Conway was made responsible for the administration of that diocese.
[20] In November 2023, he was one of 44 Church of England bishops who signed an open letter supporting the use of the Prayers of Love and Faith (i.e. blessings for same-sex couples) and called for "Guidance being issued without delay that includes the removal of all restrictions on clergy entering same-sex civil marriages, and on bishops ordaining and licensing such clergy".