After playing youth football with Manchester City and Stoke City, Broome signed for AFC Wimbledon in January 2022.[2][3] Broome made his professional debut on 9 August 2022, in a 2–0 EFL Cup defeat to Gillingham.[4] He made his league debut on 28 March 2023, keeping a clean sheet in a 2–0 victory over Walsall.[5]
Broome's contract with Wimbledon was terminated by mutual consent on 21 August 2023.[6]
On 30 August 2023, Broome signed a three-year contract for Swansea City.[7][8]