Cope was born 26 November 1907 in Surry Hills, New South Wales. He was the youngest of five sons born to Martha (née Ellem) and George Eugene Cope. His father worked as a compositor and was a "strong supporter of the Australian Labor Party".[1]
Cope attended Crown Street Public School and Bourke Street Public School. He left school after two years of secondary education and began working as a messenger boy. He later worked as a machinist at the Randwick Tramway Workshops, but was retrenched during the Great Depression and relied on the dole for three years. He also worked part-time as a cricket umpire and billiard marker. During World War II, Cope worked for Amalgamated Wireless Valve making glass tubes for use in radar. After the war's end, he manufactured glass for laboratories. He served as federal treasurer of the Australian Glass Workers' Union from 1952 to 1955.[1]
Following the death of the incumbent Labor MP Tom Sheehan, Cope was elected to the House of Representatives at the 1955 Cook by-election. His seat was abolished in a redistribution, but he switched to Watson at the 1955 federal election. He held Watson until its abolition in 1969, when he transferred to the new seat of Sydney.[1]