Allgood was born in Dublin, Ireland. Her sister was actress Maire O’Neill.  Allgood began her acting career at the Abbey Theatre and was in the opening of the Irish National Theatre Society, appearing in many of their plays all over Britain. She was frequently featured in early Hitchcock films, such as Blackmail (1929), Juno and the Paycock (1930) and Sabotage (1936).

Allgood was nominated for a Best Supporting Actress Academy Award in 1941 for her role as Beth Morgan in the 1941 film How Green Was My Valley, but lost to Mary Astor. She also had memorable roles in the 1941 retelling of Dr. Jekyll and Mr. Hyde, It Happened in Flatbush (1942), Jane Eyre (1943), The Keys of the Kingdom (1944), The Spiral Staircase (1946), The Fabulous Dorseys (1947) and the original Cheaper by the Dozen (1950).

She was married for a short time to British stage actor Gerald Henson. Both Henson and their newborn daughter died of influenza in the outbreak of 1918.  After becoming a United States citizen in 1945, Allgood died of a heart attack in 1950 at the age of 70, in Woodland Hills, California.