Weeks was born in Cardiff, Wales, to Robin and Susan (née Wade) (who have since divorced), and grew up in Chichester and Petworth, both in West Sussex. Her parents named her after honeysuckle flowers because they were in bloom when she was born. She has a younger sister and brother, Perdita and Rollo, respectively, both of whom have also pursued careers in acting.
Weeks was educated at Great Ballard School, Sussex, Roedean School and Pembroke College, Oxford, where she read English (graduating with upper-second class honours).She also spent time studying art in Italy. As a child she was a member of the Chichester Festival Theatre.
She was engaged to the poet and musician Anno Birkin for a short period before his death, aged 20, in a car crash in Italy in 2001.
Her acting career started with the juvenile lead in a television series (an adaptation of Anne Fine's Goggle-Eyes, 1993, alongside Perdita); since then she has appeared in many programmes, including the children's series The Wild House and the long-running series Midsomer Murders and Poirot. In 1997 Honeysuckle and Perdita were both in Catherine Cookson's The Rag Nymph, wherein Perdita played the younger version of her sister's character. Her film roles include Anne Ridd in Lorna Doone (2000) and Sarah in My Brother Tom (2001). She also starred in The Bill in 2008 as Julie Nowak.
Weeks 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). In the last, a BAFTA award-winning detective series set inHastings 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.