Billson was born at Ulverscroft, Leicestershire. She was educated at Bedales School, where she became Head Girl, and studied medicine at University College, London from 1918, later moving to Charing Cross Hospital due to the hostility to female students she experienced at UCL. In 1923 she married the actor Miles Malleson; they divorced in 1940. She qualified in 1926 and worked for Holborn Borough Council and the West End Hospital for Nervous Diseases, developing an interest in the fields of fertility, reproduction and sexuality.
In 1931, while working for Ealing Borough Council, she became one of the first British doctors to provide birth control advice on behalf of a local authority. In 1935 she published The Principles of Contraception, a practical guide. She became a member of the executive committee of the National Birth Control Association (later the Family Planning Association).  In 1936 the Abortion Law Reform Association was started by Alice Jenkins and Janet Chance. Malleson was amongst its more influential supporters.
She courted controversy by supporting the campaign to reform the abortion law. In 1938 she precipitated one of the most influential cases in British abortion law when she referred a pregnant fourteen-year-old rape victim to gynaecologist Aleck Bourne. He performed an abortion, then illegal, and was put on trial for it. Malleson gave evidence at the trial and Bourne's acquittal set a precedent that doctors could not be prosecuted for performing an abortion in similar circumstances. Malleson was also a supporter of eugenics and member of the Eugenics Society. In 1950 she was appointed head of the contraceptive clinic at University College Hospital.
- Leathard, Audrey (1980). The Fight for Family Planning: The Development of Family Planning Services in Britain 1921-74. Macmillan Publishers Ltd.
- Stephen Brooke, ‘Jenkins , Alice Brook (1886–1967)’, Oxford Dictionary of National Biography, Oxford University Press, May 2006; online edn, May 2008 accessed 24 Oct 2017
- D. E. Martin, ‘Malleson , Joan Graeme (1899–1956)’, Oxford Dictionary of National Biography, Oxford University Press, 2004 accessed 24 Oct 2017