On 9th August 2020 Norwich announced that six Academy youngsters, including Hills had signed with the Club as first-year scholars with the Canaries. Bradley is described as a central defender.

He played for Norwich's first-team in their pre-season friendly with Hibernian in July 2022.

He was on the bench for Norwich's Championship clash with Burnley on 25th October 2022. He then signed a new contract on 10th November 2022 which will keep him at Carrow Road until 2025 with an option of a further year.

On 17th July 2023 he joined Accrington Stanley on a season-long loan. He said "I need to be tested. A loan this season is perfect. I'm buzzing to come over. I need to use this as a stepping stone to prove to the manager that I can play for Norwich one day."

Hills made his league debut as Accrington beat Newport 3-0 on the opening day of the 2023/24 season (5th August 2023) with Hills named as man of the match. He made 48 appearances for Stanley and was named their Player of the Year.

On 24th May 2024, Hills was called up to the England Men's Elite Squad (formerly known as the Under-20s) for friendlies with Sweden and the Republic of Ireland in June 2024. Teammates Abu Kamara and Kellen Fisher were also called up as was former Canary Alex Matos. Hills came on for Blackburn Rovers youngster Ben Chrisene on 59 minutes in the game with Sweden on 7th June 2024 which England won 2-1.

After an impressive pre-season, Hills made his Norwich debut in a young back four alongside Jaden Warner, on 13th August 2024 against Stevenage in the League Cup but unfortunately he scored an own-goal. Thankfully Norwich won 4-3. (It has to be reminded that Steve Bruce also scored an own goal on his Norwich debut). His league debut came on 26th November 2024 as a late substitute in the 6-1 win over Plymouth at Carrow Road.