Born in Belfast, Agnew studied at Belfast Municipal College of Technology, Queen's University Belfast and Trinity College Dublin. He began working as a draughtsman aged 15, but soon moved into management, then, in 1933, to the Ministry of Finance, and in 1946 to the Belfast Corporation. At the Corporation, he initially worked in estates, but then for the city's Water Commissioners, of whom he was Secretary from 1952 to 1973. He was also a Fellow of the Royal Society of Health.
On retirement, Agnew stood for the Ulster Unionist Party in the Northern Ireland Assembly election, 1973, and was elected in Belfast East. In 1975, he stood for the Northern Ireland Constitutional Convention, on this occasion for the Unionist Party of Northern Ireland, but was not successful. He also stood in the Westminster seat of Belfast East at the 1979 UK general election, but took fourth place, with only 4% of the votes cast.
- Ted Nealon, Ireland: a parliamentary directory, 1973-1974, p.189
- The Times Guide to the House of Commons: 1979, p.47
- "East Belfast 1973-82, Northern Ireland Elections