Weeks was educated at Pembroke College, Oxford, where she read English (graduating with upper-second class honours). She has recently starred in The Five, created by bestselling thriller writer Harlan Coben. She is currently best known for her parts in three television series: Close Relations (1998), Ladies & Their Gentlemen (2002–2006), and Foyle's War (2002–2010, 2013, 2015). In the last, a BAFTA Award–winning detective series set in Hastings during and just after World War II, she starred opposite Michael Kitchen. In 2007, Weeks starred in The Inspector Lynley Mysteries as Tania Thompson, a character based on the Canadian serial killer Karla Homolka. In 2008, she appeared as Harriet Pringle in the Radio 4 adaptation of Fortunes of War. In 2012, she played a small part as Mrs Beeton in an episode of the BBC educational programme The Charles Dickens Show.[12]

In 2015 she appeared as Mae Harmer in the BBC TV series Death in Paradise episode 4.8. On 9 May 2015 she portrayed a wartime letter writer at VE Day 70: A Party to Remember in Horse Guards Parade, London that was broadcast live on BBC1. She also appeared in the ITV series Lewis episode "Magnum Opus".