The youngest of three siblings, Tandy was born in Geldeston Road in Hackney, London. Her mother, Jessie Helen (née Horspool), was the head of a school for mentally handicapped children, and her father, Harry Tandy, was a travelling salesman for a rope manufacturer. Her father died when Tandy was 12, and her mother subsequently taught evening courses to earn an income. Her brother Edward was later a prisoner of war of the Japanese in the Far East. Tandy was educated at Dame Alice Owen’s School in Islington. Her birth name was Jessie Alice Tandy.  Tandy began her career at the age of 18 in London, establishing herself with performances opposite such actors as Laurence Olivier and John Gielgud. She entered films in Britain, but after her marriage to Jack Hawkins failed, she moved to the United States. In 1942, she married Hume Cronyn and over the following years played supporting roles in several Hollywood films. Tandy became a naturalized citizen of the United States in 1952.  Like so many stage actors, Tandy had a hand in radio, as well. Among other programs, she was a regular on Mandrake the Magician (as Princess Nada), and then with husband Hume Cronyn in The Marriage which ran on radio from 1953 to 1954, and then segued onto television.

She made her American film debut in The Seventh Cross (1944). She also appeared in The Valley of Decision (1945), The Green Years (1946, as Cronyn’s daughter), Dragonwyck (1946) starring Gene Tierney and Vincent Price and Forever Amber (1947). Over the next three decades, her film career continued sporadically, including The Desert Fox: The Story of Rommel (1951) opposite James Mason, The Light in the Forest (1958), and a role as a domineering mother in Alfred Hitchcock’s film, The Birds (1963).  On Broadway, she won a Tony Award for her performance as Blanche Dubois in the original Broadway production of A Streetcar Named Desire in 1948. After this (she lost the film role to actress Vivien Leigh), she concentrated on the stage. In 1977, she earned her second Tony Award, for her performance (with Cronyn) in The Gin Game and her third Tony in 1982 for her performance, again with Cronyn, in Foxfire (play).  The beginning of the 1980s saw a resurgence in her film career, with character roles in The World According to Garp, Best Friends, Still of the Night (all 1982) and The Bostonians (1984). She and Cronyn were now working together more regularly on stage and television, including the films Cocoon (1985), *batteries not included (1987) and Cocoon: The Return (1988) and the Emmy Award winning television film Foxfire (1987, recreating her Tony winning Broadway role). However, it was her colourful performance in Driving Miss Daisy (1989), as an aging, stubborn Southern-Jewish matron, that earned her an Oscar.  She received a Best Supporting Actress nomination for her work in the grassroots hit Fried Green Tomatoes (1991), and co-starred in The Story Lady (1991 telefilm, with daughter Tandy Cronyn), Used People (1992, as Shirley MacLaine’s mother), television film To Dance with the White Dog (1993, with Cronyn), Nobody’s Fool (1994), and Camilla (also 1994, with Cronyn). Camilla proved to be her last performance, at the age of 84.

Prior to moving to Connecticut, she lived with Cronyn for many years in nearby Pound Ridge, New York, and they remained together until her death in 1994. In 1990, she was diagnosed with ovarian cancer and fought it for the next four years. Tandy also suffered from angina and glaucoma. Despite all this and her advancing age, she continued working. She died at home on 11 September 1994 in Easton, Connecticut, at age 85.