Montagu was born in Leeds, West Riding of Yorkshire, to Lieutenant-Colonel John Drogo Montagu (1916-2013), whose great-great-grandfather Admiral George Montagu was great-great-grandson of Lord James Montagu (d. 1665), who, in his turn, was younger son of Henry Montagu, 1st Earl of Manchester. She attended Loughborough University and the Webber Douglas Academy of Dramatic Art.
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, Health and Efficiency, Wish Me Luck, Tumbledown, The Harry Enfield Television Show, 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 and The End of the F***ing World.
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.