Tennant was born in London, England. Her mother, Irina Baronova, was a Russian-born prima ballerina who appeared with the Ballet Russes de Monte Carlo, and her father, Cecil Tennant, was an English producer and talent agent who ran MCA's talent office and whose clients included Laurence Olivier, Vivien Leigh, John Gielgud and Michael Redgrave.

Her maternal grandfather, Misha Baronov, was a senior officer in the Russian navy, and her maternal grandmother, the former Lydia Vishniakova, was a general's daughter. Olivier was Tennant's godfather.

She has a sister, Irina, and a brother, Robert, both of whom were badly injured in the car accident that killed their father in 1967. She attended the Central School of Speech and Drama at the University of London.