Slovo started her career in theatre, before moving into the television and film industry, working first as a script editor and development executive for the BBC, and then as a film producer for Company Pictures and Working Title Films.
Slovo's family is Jewish. She is the daughter of Joe Slovo and Ruth First — both major figures in the anti-apartheid struggle who lived perilous lives of exile, armed resistance, and occasional imprisonment, culminating in her mother's assassination in 1982. A family memoir in the form of a feature film, A World Apart, was written by her sister Shawn Slovo and starred Barbara Hershey. She played her mother in the film Catch a Fire, also written by her sister Shawn Slovo. She is the youngest sister of novelist Gillian Slovo and screenwriter Shawn Slovo.
In December 2019, along with 42 other leading cultural figures, Slovo signed a letter endorsing the Labour Party under Jeremy Corbyn's leadership in the 2019 general election. The letter stated that "Labour's election manifesto under Jeremy Corbyn's leadership offers a transformative plan that prioritises the needs of people and the planet over private profit and the vested interests of a few."
- Dead Proud from Second Wave Young Women Playwrights - EDITED BY ANN CONSIDINE & ROBYN SLOVO
- "Vote for hope and a decent future". The Guardian. 3 December 2019. Retrieved 4 December 2019.
- Proctor, Kate (3 December 2019). "Coogan and Klein lead cultural figures backing Corbyn and Labour". The Guardian. Retrieved 4 December 2019.