Coote was born in Frodsham, Cheshire, the daughter of the painter Mark Coote. She was educated at the Guildhall School of Music and Drama in London (though she did not complete her course), the Royal Northern College of Music in Manchester (where she came into contact with Janet Baker and Brigitte Fassbaender) and the National Opera Studio during 1995/96. Coote was a BBC Radio 3 New Generation Artist from 2001 until 2003. She sings both operatic (particularly trouser roles) and recital repertoire, in the latter often with pianist Julius Drake.

An interpreter of Handel ("his music could keep you going for a whole career") she has performed contemporary pieces such as Dominick Argento's From the Diary of Virginia Woolf, a partly atonal work first performed by Janet Baker, an influence on Coote. Judith Weir has written a song cycle, The Voice of Desire, especially for her; it was premiered at a BBC Chamber Prom.

Coote has performed at England's Opera North, the English National Opera, the Metropolitan Opera in New York (Hansel in Humperdinck's Hansel and Gretel), the San Francisco Opera in 2002 (Ruggiero in Handel's Alcina) and 2008 (Idamante in Mozart's Idomeneo).[3] In 2009, she sang Maffio Orsini in Gaetano Donizetti's Lucrezia Borgia at the Bavarian State Opera. She also appeared in 2011 as Prince Charming in Cendrillon at the Royal Opera House. In 2013, she played Sextus in the Metropolitan Opera's production of Handel's Giulio Cesare. In March 2017 she reprised the role of Idamante in six performances at the Metropolitan Opera.