Weisz was born in London, England and grew up in Hampstead Garden Suburb. Her mother, Edith Ruth (née Teich), is a Vienna-born Austrian teacher turned psychotherapist. Her father, George Weisz, is a Hungarian-born inventor whose family fled to England to escape Nazi persecution. Weisz's father is Ashkenazi Jewish and her mother has been referred to as either Catholic, Jewish, or having Jewish ancestry. Weisz was raised in a cerebral Jewish household and refers to herself as Jewish. Weisz has a sister, Minnie Weisz, who is an artist.

Weisz was educated at North London Collegiate School. She was then sent to Benenden School and eventually settled in St Paul's Girls' School. She then entered Trinity Hall, Cambridge, where she graduated with a 2:1 in English. During her university years she appeared in various student productions, co-founding a student drama group called Cambridge Talking Tongues, which went on to win a Guardian Student Drama Award at the Edinburgh Festival for a