[9][3] Broome made his professional debut on 9 August 2022, in a 2–0 EFL Cup defeat to Gillingham at Plough Lane.
[12] His contract with Wimbledon was terminated by mutual consent on 21 August 2023 as manager Johnnie Jackson explained that Broome was "finding it difficult being away from home and he wanted a new challenge".
[13] On 30 August 2023, Broome signed a three-year contract with Championship club Swansea City after having impressed head of goalkeeping Martyn Margetson whilst playing for England Under-17 and Under-18.
[16] On 14 January 2025, Broome joined League Two club Port Vale on loan for the remainder of the 2024–25 season as manager Darren Moore lost regular custodian Connor Ripley to paternity leave.
[17][18] He made his debut for the Valiants on 4 February, conceding four goals in a 4–1 defeat to Wrexham in an EFL Trophy match at Vale Park, though Moore blamed the defence for leaving him exposed.