Connolly was born in County Durham in North East England on June 13, 1963, and grew up in Middleton St. George. She sang in a village choir and studied piano and cello as a youngster in addition to singing. At Queen Margaret's Boarding School, she continued with piano and was impressed by her sister's performance in Purcell's Dido and Aeneas, but continued to focus on piano at the Clarendon College of Further Education (as well as winning college championships in tennis and table tennis). Connolly even earned a piano education degree at the Royal College of Music before switching definitively to singing. After graduating, she joined the BBC Singers and became a freelance choral singer, flirting as well with a jazz career. A second-place finish at the International Vocal Competition s'Hertogenbosch, Opera Section, as well as meeting teacher Gerald Martin Moore confirmed her choice of classical music, and she landed solos in Bach concerts conducted by Philippe Herreweghe in the 1990s. In 1997, she made her recording debut on an album of Bach Advent Cantatas conducted by Herreweghe, with his Collegium Vocale, on the Harmonia Mundi label.
She continued to make inroads into the early music scene, recording Rameau's Les Fêtes d'Hébé with William Christie's Les Arts Florissants and performing the title role in Handel's Xerxes in several houses, including the New York City Opera. Gaining a reputation as a Handel specialist, she has moved into Romantic and contemporary repertory as well, performing heavy works such as Mahler's Rückert-Lieder and Elgar's The Dream of Gerontius; in 2011, as part of the observances of the centenary of Mahler's death, she performed all his vocal works with the Philharmonia Orchestra. Connolly has also supported contemporary music, performing and recording Mark-Anthony Turnage's Twice Through the Heart.
Connolly's extensive recording career includes many song recitals, both orchestral and voice-and-piano, that were original in conception. These included The Exquisite Hour (2006), which inaugurated her ongoing relationship with the Signum label. Connolly appeared with the choir Tenebrae on the 2017 release A Walk with Ivor Gurney. That year, she was made a Dame Commander of the Order of the British Empire. Connolly also recorded for Chandos, and for that label, she appeared on collections of pieces by Elgar, Arthur Bliss, and Frank Bridge between 2018 and 2022. In 2022, she released a complete recording of the songs of Henri Duparc on Signum Classics. ~ James Manheim, Rovi