Harvey began his career at Portsmouth before being signed by Watford in 2017. Whilst there, he had loan spells at St Albans City and Hungerford Town. He then joined Oxford United and returned to Hungerford Town on loan.
It was after a further loan spell at Woking when Bradbury finally made his debut for Oxford. He went on to rejoin Woking and helped the club to win promotion out of the National League South via the play-offs at the expense of the Wings. He signed with Millwall in July 2019 and spent the second half of the 2019–20 season on loan at Morecambe. He signed for Oxford City for the 2020/21 season but arrived at Welling in March 2022.