Beaumont began her career as an actress, before moving to theatre production and stage management. In 1964, she was hired as an editor for the BBC. She transitioned in the company to directing and production management, before leaving the company to pursue film production. In 1971, she made her directorial debut with the horror films The Johnstown Monster and Crucible of Horror. From 1973 to 1980, she worked for Thames Television where she directed daytime television programs.
In 1980, Beaumont produced and directed the horror film The Godsend. After its American release, Beaumont was able to secure a meeting with American television producer Aaron Spelling. Following the meeting, she moved to America and began a decades-long working relationship with Spelling, who hired her to direct shows such as Vega$, Hart to Hart, Glitter, and Beverly Hills, 90210. While working with Spelling on Dynasty, she recommended that he cast her friend Joan Collins as Alexis Colby.
When Beaumont first moved to California, she was one of less than 100 professional female directors working in the film and television industry. Despite this, Beaumont stated that the US provided more opportunities for women than the UK, choosing to stay in America for the remainder of her directorial career. In 1986, she was nominated for a Primetime Emmy for her direction on the set of Hill Street Blues. In 1989, she became the first woman to direct an episode of Star Trek, working on the Star Trek: The Next Generation episode "Booby Trap".