Graveson began performing on the club circuit at the age of six, but made her television debut in White Peak Farm in 1988.

Graveson was seen in an episode of the BBC hospital drama Holby City in 2006, and in 2007 she appeared in the ITV sitcom Benidorm.

The film is largely a succession of dialogues between the two, intermittently broken up by events such as Richard hiding in the wardrobe to watch Naomi at work, or a backstory involving his wife and daughter, which gradually becomes more and more important.

Her Indian work includes Khufiya, Gully Boy, Made In Heaven, Dr. Max (Netflix) and Shoojit Sircar (a PayPal commercial).

She has also appeared in Hard Times at the Piccadilly Theatre in London's West End, and in Raving Beauties with John McArdle at the Liverpool Playhouse (1991–1992).

In 1993–94, Graveson earned a Tony Award nomination in the "Best Performance by a Featured Actress in a Musical" category, for her role as Linda in Blood Brothers N.Y.C.

Graveson started her professional stage career at the Live Theatre, Newcastle in 1983 alongside Robson Green.