Burnell's mother is a dance teacher, and her father is a telecoms manager. She has one younger brother. She was originally named "Claire" but started asking people to call her "Cerrie" at the age of 10. Burnell grew up in the south-east London suburb of Petts Wood.
Burnell was born with her right arm ending slightly below the elbow. Her parents encouraged her to wear a prosthetic arm, but she resisted from the start and stopped wearing one entirely when she was nine. Burnell also suffered from dyslexia, which left her unable to read until the age of 10. She learned with extra tuition and the Letterland system.
Burnell has a daughter, Amelie, born in 2008.