Swinburne was born in Bath, Somerset, the daughter of Henry Swinburne Johnson and his wife Leonora Tamar (née Brain). She was educated at Rosholme College, Weston-super-Mare, and studied for the stage at the Royal Academy of Dramatic Art. As a member of Clive Currie's Young Players in 1914, she appeared at the Grand, Croydon, Court and Little Theatres, during that year.
In 1914 she attended an audition with the ballerina Phyllis Bedells and later Anna Pavlova who considered her too young, even if very talented, for the corps de ballet. Nora instead joined the Italia Conti school where she obtained her first real part as a child actress in Where the Rainbow Ends. She performed in the show in London and in all the big cities of Britain for eighteen shillings (90p) a week.