Born in Liverpool on 19 August 1914, Heilbron was educated at the Girls' Public Day School Trust's Belvedere School, Liverpool. She later studied at the University of Liverpool, graduating LLB with First Class Honours in 1935 and LLM in 1937. She was called to the bar in 1939 and was elected to the northern circuit in 1940.

She married Nathaniel Burstein in 1945; their daughter Hilary was born in 1949. Also in 1949 she became one of the first women to be appointed king's counsel in England and Wales.

She built a reputation as a highly successful defense advocate, working on such cases as the Cameo murder case of 1950 and the anti-discrimination case brought by cricketer Learie Constantine in 1944. In 1972 she became the first woman to sit as a judge at the Old Bailey. In 1974 she was appointed to the High Court Bench (becoming a Dame), retiring from this position in 1988.

She died on 8 December 2005.