She was born in Hull, East Yorkshire, to Herbert Rix of the family shipbuilding company JR Rix & Sons, and his wife, Fanny (nee Nicholson), an amateur actor and singer. When Sheila was young she saw more of her nanny than her parents, and was educated at the Canonesses of St Augustine school in the city and then St Ethelburga's in Hornsea, where she began acting in school plays and performing in concerts put on by her mother.