Conway became a member of the Irish Privy Council in 1660 and was a confidant of James Butler, 1st Duke of Ormonde.

He became a Fellow of the Royal Society on 2 January 1668, became governor of Charlemont Fort in 1671 and served as Master of the Ordnance in Ireland from 1679.

On 2 February 1681 he joined the Privy Council of King Charles II and became Secretary of State for the Northern Department.

He resigned in January 1683 amid allegations that he was complicit in "crimes and misdemeanours ... either in relation to the King's person or his public negotiations or transactions with foreign ambassadors, or in not rightly pursuing the King's instructions to ambassadors abroad".

He married three times, but had no children who survived infancy: He died on 11 August 1683 and was buried in Arrow, Warwickshire.