Tanner was born in 1925 and studied at the Royal Academy of Dramatic Art (RADA). She left her focus on dramatic acting to form the Tanner Sisters with her sister, Frances Tanner. Together the Tanner Sisters made hundreds of appearances on British radio and television during the 1950s, even performing alongside Buddy Holly in March 1958. The Tanner Sisters often played with the Hedley Ward Trio, appearing with them first on the Educating Archie radio show from 1950 to 1952.
Tanner pursued a career in television comedy after she and Frances ended the Tanner Sisters act after ten years. Tanner made her television acting debut in 1960 on Sykes and A..., Opposite Eric Sykes. She was cast in guest roles in numerous sitcoms, including an episode of The Rag Trade in 1963, Fawlty Towers in 1979, and It Ain't Half Hot Mum in 1981. From 1971 to 1972 she portrayed Adam Faith's girlfriend's mother in the ITV series, Budgie. Tanner was also cast as Luisa di Marco in EastEnders throughout 1998, as well as a role in Coronation Street.
Stella Tanner died on 26 March 2012 at the age of 87. Her daughter, Alexa survived her. Tanner's husband, the American actor, David Bauer, died in 1973.