Sawalha was born in London, the daughter of Roberta and Nadim Sawalha. She was named after her grandmother, a Jordanian businesswoman who had received an award from Queen Noor for enterprise. She is of Jordanian, British, and French Huguenot ancestry.

She was born into an acting family: her father Nadim is an actor who appeared in the James Bond movie The Spy Who Loved Me, while her sister Nadia starred in the soap EastEnders and is now a television presenter and chat show host.

She first gained public attention for her starring role in the ITV teenage comedy/drama Press Gang, which ran from 1989 to 1993. The on-screen chemistry between the two leads was reflected off-screen, as she and co-star Dexter Fletcher became an item for several years.

From 1991-94, she starred in the ITV family comedy Second Thoughts and continued with her character, Hannah (Lynda Bellingham's daughter), in the British Comedy Award winning Faith in the Future (1995–98).

From 1992 to 2005, she played Saffron Monsoon in the sitcom Absolutely Fabulous alongside Jennifer Saunders.

She starred in the 1995 BBC adaptation of Jane Austen's Pride and Prejudice as Lydia Bennett, co-starring opposite Jennifer Ehle and Colin Firth. She also voiced Ginger in DreamWorks/Aardman's Chicken Run.

In 2000 she appeared as Janet, the Australian Barmaid ("Built for bar work; it's instinct... instinct!!") in the first series of the British sitcom Time Gentlemen Please. She also played the much put-upon PA to "Zak" in Argos TV adverts during 2002-2004, along with Richard E. Grant. She has also joined acclaimed actor Ioan Gruffudd in the internationally successful TV/DVD adaptations of CS Forester's Horatio Hornblower novels, as the captain's first wife Maria.

In 2001, she became the Alan Davies co-star in Jonathan Creek after Caroline Quentin left, appearing in a Christmas Special ("Satan's Chimney"). She returned for a series between 2003-2004.

In 2006, she participated in the third series of the genealogy documentary series Who Do You Think You Are? tracing her family's roots, which are Jordanian Bedouin on her father's side, and French Huguenot on her mother's. She also appeared in the pilot of BBC 1's "A Taste of my Life" presented by Nigel Slater.

After a two year break, she was back on screen in May 2007, competing in the BBC dog training celebrity reality show The Underdog Show.