Donaldson was born in 1948 and brought up in Hampstead, London with her younger sister, Mary. The family lived in a Victorian three-storey house near to Hampstead Heath where the sisters liked to play. Her parents, sister, and their pet cat named Geoffrey, lived on the ground floor; while an aunt and uncle lived on the first floor, and her granny lived on the second floor.

Donaldson's father graduated in Philosophy, Politics and Economics from the University of Oxford before World War II, and worked as a musician playing the cello. He contracted polio when Donaldson was six years old and he was initially hospitalised. His attempts to walk with crutches and callipers were unsuccessful, so he had to resort to using a wheel chair. This was at a time when there were not many aids and facilities for the handicapped in public places, but he was able to continue working with help from his wife who assisted with some arduous physical tasks such as transfers between a car and wheelchair.

Donaldson studied Drama and French at the University of Bristol, and as a student she used to busk with a female friend to earn extra money for educational trips to Paris. The duo were subsequently joined by their friend Malcolm, a medical student who played a left-handed guitar and sang, and who would later become Donaldson's husband.

Donaldson's husband is a consultant paediatrician working in Glasgow, Scotland, and the couple live in Bearsden, East Dunbartonshire. Their first child, Hamish, was diagnosed with a schizoaffective disorder when he was about 17 years old having had a psychotic episode. Hamish committed suicide in 2003, aged 23, at a time when he was drinking alcohol heavily. Their other two sons are university students. Her father died of a heart attack at 59.

Donaldson is a patron of ArtLink Central, a charity which helps artists to work with disadvantaged people.