Her parents were Scottish and emigrated to South Africa where she was brought up. Her father was a doctor, a neurologist who was in the British 14th Army in Burma in the Royal Army Medical Corps. She attended the Herschel Girls School, an independent boarding school in Cape Town. She completed her education at the Ecole de Commerce in Geneva and at an English college called the House of Citizenship.
For a time in London, she worked as a typist at Australia House, then became a temporary junior secretary at the BBC. This entitled her to an induction course, where she was taught the BBC's method of working.
Returning to South Africa she began her broadcasting career there on the SABC's English language radio service. Initially joining the BBC in 1967 as a reporter for The World At One, she went on to host Woman's Hour from 1972 until 1987 and beforehand read the news for BBC Radio One in 1971. In 1984 she became one of the hosts of BBC Radio 4's Today programme, a position she held until 2002.
That same year she was appointed a CBE for her services to broadcasting. Sue is a trustee of UNICEF and also of the John Ellerman Foundation. She is also on the Chancellor's Forum for the London Institute and an honorary graduate of several universities, among them Nottingham, Nottingham Trent and Dundee.
She hosts The Reunion on BBC Radio 4.
In late 2010, she recorded her final episode as chair of the popular book programme A Good Read, also on BBC Radio 4. At seven years and approximately 500 book reviews, she is the show's longest serving presenter.
Her autobiography, Woman of Today, published in 2002, contained candid revelations and outlined her claimed relationships, including an alleged affair with Leonard Rossiter (by then deceased and unable to confirm or deny the claim).