Montagu married Alan Nixon in 1984. They divorced in 2006. The couple have two children, Olivia (born 1990) and Luke (born 10 Aug 1996). Felicity and Olivia worked together in 2016 in the BBC Radio 4 series Guilt Trip, where they played a fictional mother and daughter.

In 2004 and 2005, Montagu starred in ITV comedy drama Doc Martin as Caroline Bosman, the radio presenter of Radio Portwenn. She co-starred as Lynn, the faithful but put-upon personal assistant, in I’m Alan Partridge and as the huge-breasted, raunchy vicar’s wife Sue 2 in Nighty Night. In 2006, she took the leading role of housewife and gang queen Barbara Du Prez in the offbeat comedy series Suburban Shootout. Her other television credits include Mapp and Lucia, The Bill (‘Furthers’; 1991) Health and Efficiency, Wish Me Luck, Tumbledown, Harry Enfield’s Television Programme, 2point4 Children, A Touch of Frost, My Life as a Popat, Johnny and the Bomb, Peak Practice, The Queen’s Sister, Coogan’s Run, Doc Martin, Lead Balloon, Alexei Sayle’s Stuff, Skins, Hank Zipzer, M.I. High, The End of the F***ing World, and Sally4Ever. In 2020, she appeared as a contestant on Celebrity Masterchef.

Montagu’s reputation in comedy character parts was enhanced by her performance in Bridget Jones’s Diary as Perpetua, Bridget’s unpleasant colleague. She also appeared in the 2006 film Confetti as highly strung magazine editor Vivien Kay-Wylie. She appeared in the film I Want Candy in which she plays the mother of an ambitious teenager. She appeared in How to Lose Friends & Alienate People (2008). In 2013, she reprised her role as Lynn Benfield for the Alan Partridge film Alan Partridge: Alpha Papa. and Mrs Mainwaring in the 2016 Dad’s Army film. In 2016, Montagu reprised her role as Miss Adolf from the Hank Zipzer series in the TV movie Hank Zipzer’s Christmas Catastrophe.

After graduating from drama school Montagu was cast in the Terry Johnson play Unsuitable for Adults at the Bush Theatre. She went on to work at the Royal Court Theatre and the Royal National Theatre. After a long hiatus due to her screen work, Montagu returned to the stage when she starred as Melanie alongside Rowan Atkinson in the 2013 West End production of Quartermaine’s Terms at Wyndham’s Theatre. In 2017 she appeared at the Liverpool Playhouse in a stage adaptation of Jane Austen’s Pride and Prejudice as Mrs Bennet alongside Matthew Kelly as Mr Bennet.