[6] Bush worked for the magazine Progress (linked to the organisation of the same name) before writing for The Daily Telegraph, including working on the Morning Briefing email as editorial assistant to Benedict Brogan.

[7][8] Later that year, he was the first political commentator to predict Jeremy Corbyn's election as Labour leader after obtaining leaked internal poll data.

[7][9] He has appeared on the BBC's current affairs programme Newsnight, commenting on UK politics,[10] and was tipped as an outsider to succeed Nick Robinson as the BBC's political editor in 2015, a position that eventually went to Laura Kuenssberg.

[14] In December 2018 he was appointed political editor of the New Statesman, while also writing a fortnightly column for the i newspaper.

[17][18] On 8 June 2020 he was appointed to chair the Board of Deputies of British Jews' Commission on Racial Inclusivity in the Jewish Community.