He is best remembered for having played the lead role of bus driver Stan Butler in the LWT sitcom On the Buses (1969–1973) and its three spin-off feature films.

Having performed as a music hall entertainer, Varney first came to national recognition as factory foreman Reg Turner in the BBC sitcom The Rag Trade (1961–1963).

He also played in working men's clubs, pubs and ABC cinemas with his friend George Shears and later sang with big bands of the time.

During the Second World War, Varney joined the Royal Engineers, but continued his performing career as an army entertainer, touring in the Far East for a time.

After being demobilised in the late 1940s, he starred on stage in a comic revue entitled Gaytime, with Benny Hill as his partner in a double act.

Slightly later, he starred in a children's show for BBC TV called The Valiant Varneys (1964–65), performing multiple characters in front of a live audience.

Varney featured in The Great St Trinian's Train Robbery (1966) with Frankie Howerd, Dora Bryan and George Cole.

Varney took considerable lengths to prepare for the role, even attempting to gain a public service vehicle licence so that he could be filmed driving on the open road.

Ultimately, he only appeared in one further non-Buses film, The Best Pair of Legs in the Business (1973), and two television series, both made by ATV for the ITV network: an eponymously titled sketch show (1973–74) and another sitcom, Down the 'Gate (1975–76), which was set in Billingsgate Fish Market.