Drew was born in Gloucestershire, the son of an accountant, and was educated at the Kingsfield School, Kingswood before attending the University of Nottingham where he was awarded a Bachelor of Arts degree in economics in 1974 and the University of Birmingham where he qualified as a teacher and received his Postgraduate Certificate in Education in 1976. He went on to study for his master's degree at the Bristol Polytechnic, where he graduated with an MA in historical studies in 1988. He was awarded a Master of Education from the University of the West of England in 1994.
He began his professional career in education as a teacher at the Princethorpe College in Rugby, Warwickshire in 1976, moving in 1978 to teach at the St Michael's School in Stevenage, Hertfordshire. In 1982 he moved back to his native Gloucestershire when he went to teach at the Maidenhill School in Stonehouse, before moving to the Dene Magna Community School in Mitcheldean in the Forest of Dean. Throughout his teaching career he was a member of the National Association of Schoolmasters Union of Women Teachers, and was a branch secretary 1984-1986. In 1986 he left teaching at school to become a senior lecturer in education at the University of West England, where he remained until his election to Westminster in 1997, and remains a member of the National Association of Teachers in Further and Higher Education.
He was elected as a councillor for the Stevenage Borough Council in Hertfordshire for a year in 1981, and was elected as a councillor for the Stroud District Council in 1987, where he remained until 1995. He was also elected to the Stonehouse Town Council in 1987 and remains on the council. To make the collection of tiers of local government complete, he was elected as a councillor on the Gloucestershire County Council in 1992, stepping down on his election to Parliament. He was elected as the secretary to the Stroud Constituency Labour Party for a year in 1992. He has also been a member of UNISON since 1990.
He unsuccessfully contested Stroud at the 1992 general election, but was defeated by the incumbent Conservative, Roger Knapman by 13,405 votes. He succeeded in taking the seat from Knapman at the 1997 general election with a majority of his own of 2,910, and remained the MP until 2010. He made his maiden speech on 17 June 1997. He successfully defended his majority in 2001 general election, and retained it, but with a significantly reduced majority of just 350 in the 2005 general election. In Parliament, he was a member of the Agriculture Select Committee, and its successor Environment, Food and Rural Affairs Committee from 1999 until 2010.
He was identified by The Herald newspaper in Scotland following the 2005 election as one of the more frequent Labour backbench rebels, having 'rebelled' over both the Iraq War and terror legislation during the 2001 parliament. His staunch Eurosceptic stance was also at odds with the Europhile views of the leading members of the Labour Government.
A member of the Socialist Campaign Group, he nominated John McDonnell in the 2007 Labour leadership election.
During the 2010 election, he lost out by 1,299 votes (2.0%) to the Conservative Neil Carmichael who won with 40.8% of the vote, with Liberal Democrat Dennis Andrewartha taking 15.4%. Despite the loss, he actually managed to increase the number of votes he won compared to the 2005 election, as well as managing to obtain the smallest losing Labour swing in the whole of England.
In the 2011 Local Elections, he returned to Stroud District Council after being elected to the Farmhill and Paganhill seat, taking 63.9% of the vote.
He stood as the Labour Co-operative candidate for Stroud at the 2015 general election, but failed to take the seat. He stood again at the 2017 general election and regained the seat with a majority of 687 over the Conservatives.
Following his return to the Commons he was appointed as Shadow Farming and Rural Affairs Minister on 3 July 2017 by Labour leader Jeremy Corbyn.
Drew has been married to his second wife Anne Baker since 1990 and has two daughters and two sons. They live in Stonehouse. He has been a vegetarian for 25 years.
He was appointed Chairman of Forest Green Rovers in May 2010, and later became Vice Chairman. He resigned from the role upon his re-election to the Commons in June 2017.