Wright performed in school plays, but credits her desire to be a professional actress to the 2006 film Akeelah and the Bee. She found Keke Palmer's performance inspiring, remarking that the role "resonated. It's one of the reasons why I'm here". She attended the Identity School of Acting, enrolling at the age of 16. She appeared in two episodes of Holby City and Top Boy in 2011. She had a small role in My Brother the Devil in 2012, where she was recognized by Screen International as one of its 2012 Stars of Tomorrow. Michael Caton-Jones cast Wright in her first leading role in Urban Hymn (2015), which brought her to the attention of Hollywood. The same year, she appeared in an episode of Doctor Who, and the following year, she began a recurring role as Renie on Humans. During this time, she also appeared in the play Eclipsed in the London's Gate Theatre. In 2017, Wright starred in the Black Mirror episode "Black Museum"; her performance earned her a Primetime Emmy Award nomination for Outstanding Supporting Actress in a Limited Series or Movie.