She attended Blanchelande College, a local independent school for girls, and she went on to study biology at the University of Bristol and gained a Bachelor of Science
Montague"s first job was as a stockbroker for County National Westminster, and then a Eurobond dealer with National Westminster Capital Markets in London. She then went into business in London with the owner of Charles Tyrwhitt Shirts. Montague began her journalistic career with Channel Television in 1991.
She joined Reuters in January 1995, and then became business correspondent for Sky News in January 1996.
She joined the British Broadcasting Corporation during October 1997, and presented Newsnight and British Broadcasting Corporation News, before joining the Today news programme on British Broadcasting Corporation Radio 4 in 2002. She has also presented HARDtalk on British Broadcasting Corporation News.
In December, 2008 she hosted British Broadcasting Corporation World"s Nobel Minds in the library of the Royal Palace, Stockholm. The 2008 Nobel Prize winners in Physics, Chemistry, Medicine, Economics and Literature had a round-table discussion on issues of global concern and their own contributions to the world of knowledge.
On 6 November 2010 she was noted for breaking a strike at the British Broadcasting Corporation, called by the National Union of Journalists, when she arrived to present the Today programme at 3:30am, along with fellow presenter Evan Davis, thereby not technically crossing the picket line.
On 15 July 2011, she again bypassed an NUJ strike presenting Today with Justin Webb, again by arriving early. On 10 February 2015 she was awarded the degree of Doctor of Laws honoris causa from the University of Bristol.