Although most people now see her as an English actress, Glynis was born in South Africa, an only child, and grew up in Johanesburg. Her parents split up when she was a child. When Glynis was seventeen, her mother died. A year later, in 1974, with monetary support from her stepfather, she moved to England to attend the Mountview Theatre School, London. After theatre school, Glynis began to get parts in theatre (including roles with the Royal Shakespeare Company). In 1976, she married the actor Paul Antony Barber, but this did not last long. With her film and television work multiplying, they were divorced in 1979.
She married actor Michael Brandon, her former co-star in Dempsey and Makepeace, in 1989.