Drewery grew up in Nottingham and the Lincolnshire village of Authorpe and went to South Reston Primary School, then Monks' Dyke High School and King Edward VI Grammar School in nearby Louth, then Lincoln College. She grew up on the classic pop standards, since her father played in a band which was regular support to big stars like Tom Jones and Sandie Shaw. She was strongly influenced by Northern soul, visiting performances at the Winter Gardens in Cleethorpes, and referred to Northern soul tracks during an extensive interview on BBC Radio Nottingham. Her mother looks after Authorpe Hedgehog Care.
Drewery moved to London in 1976 to study fashion at the Central Saint Martins College of Art and Design and eventually became a fashion designer. But she also sang for short periods with bands like Working Week.
In 1984 she first met with Andy Connell and Martin Jackson and together they formed Swing Out Sister. Apart from singing the lead vocals in the band, she is the band's lyricist, and participates in the composing of the songs together with her musical partner Connell. The group's most recent album, Beautiful Mess, was released in the U.K. and Japan in 2008 and in the U.S. in 2009.