Pugh was born on 3 January 1996 in Oxford. Her father, Clinton Pugh, is a restaurateur in Oxford, while her mother, Deborah, is a dancer and dance teacher. Pugh has three siblings: actor and musician Toby Sebastian, actress Arabella Gibbins, and youngest sibling, Rafaela Pugh.
Pugh made her American television debut in the television film Studio City, co-starring Eric McCormack, in 2015. The next year, she starred in the independent drama Lady Macbeth and had a recurring role as a webcam model in the first season of the ITV detective series Marcella. In the former, based on the novella Lady Macbeth of the Mtsensk District by Nikolai Leskov, she played a young girl unhappily married to a much older man. Reviewing the film for Variety, Guy Lodge called Pugh a "a major talent to watch" and praised her portrayal of her character's "complex, under-the-skin transformation". For her performance, she won the BIFA Award for Best Performance by an Actress in a British Independent Film, among others.
In 2018, Pugh appeared in the action film The Commuter and played Cordelia to Anthony Hopkins's Leir of Britain in Richard Eyre's television film King Lear. Later that year, she portrayed Elizabeth de Burgh in the Netflix historical film Outlaw King (2018), which also stars Chris Pine as Robert the Bruce. Charles Bramesco of The Guardian found her "excellent despite her thankless role". She next appeared in a six-part miniseries adaptation of John le Carré's spy novel The Little Drummer Girl, in which she played an actress in the 1970s who becomes embroiled in an espionage plot. In a mixed review of the series, Richard Lawson of Vanity Fair wrote, "Pugh is terrific throughout, once again asserting her star-on-the-rise status. She smartly mixes earthiness with sophistication, wisdom with naïveté."
Pugh was listed on Forbes' annual 30 Under 30 list, which recognises the 30 most influential people in Europe under the age of 30, in 2019. She was also recognised as having a breakthrough in the same year, during which she starred in three major films. She first starred as professional wrestler Paige in Fighting with My Family, a comedy-drama about Paige's relationship with her family, co-starring Lena Headey and Dwayne Johnson. The film premiered at the 2019 Sundance Film Festival to critical acclaim. Geoffrey Macnab of The Independent noted how different the role was from Pugh's previous appearances, and wrote that she was "completely convincing as the wrestler" and that she showed "the same defiance, scruffy glamour and self-deprecating humour as the real life [...] Paige." She next played the lead role in Ari Aster's horror film Midsommar, which chronicles a troubled couple who encounter Swedish cultists. David Edelstein of Vulture called her performance "amazingly vivid" and wrote, "Her face is so wide and open that she seems to have nowhere to hide her emotions."
Since April 2019, Pugh has been in a relationship with American actor Zach Braff. As of January 2020, she resides in London.