Baron Plumb's family were from Cheshire and had been in farming for several generations. Plumb joined his father in 1940, then he took over running the farm after his father’s death. In 1965 he went into politics as a leader of the National Farmers Union.
Between July 1979 and 1999 he was member of the European Parliament. During this period, he was member and chair of the European Democratic Group; member, vice-chair and member of the Bureau of the Group of the European People's Party (Christian-Democratic Group). Plumb was Chair of the Committee on Agriculture, Chair of the Members from the European Parliament to the Joint Assembly of the Agreement between the African, Caribbean and Pacific States and the European Union (ACP-EU), member of the Committee on Development and Cooperation, member of the Committee on Agriculture, Fisheries and Food, member of the Committee on Transports, member of the Delegation for relations with the countries of South America, member of the Committee on Agriculture, Fisheries and Rural Development, member of the Delegation for relations with Hungary, member of the Delegation to the EU-Hungary Joint Parliamentary Committee, member of the Members from the European Parliament to the Joint Assembly of the Agreement between the African, Caribbean and Pacific States and the European Union (ACP-EU), member of the Temporary committee of inquiry into BSE (bovine spongiform encephalopathy). Between January 1987 and July 1989, Plumb was President of the Parliament's Bureau and President of the European Parliament.
Plumb was made a Life peer as Baron Plumb, of Coleshill in the County of Warwickshire, on 6 April 1987. He retired from the House of Lords on 3 November 2017.