Roberts was born in Barnsley, South Yorkshire, and raised in Penistone, where he attended Penistone Grammar School. He played football for Barnsley Boys and for Barnsley College, and as a 17-year-old was offered a trial with Barnsley F.C. but a broken ankle prevented his taking part. Roberts began his senior football career with Wakefield of the Northern Premier League Division One North in 2008. After 17 first-team appearances, he had a trial with then Championship club Crystal Palace, on the recommendation of Wakefield manager Ronnie Glavin. No deal ensued, and Roberts returned to Wakefield to complete a season in which he won the Directors' Player of the Season award. In 2009–10, he won both Players' and Supporters' Player of the Season awards. In both campaigns, he won more man-of-the-match awards than any other Wakefield player. In 2011, he moved up a division, signing for Worksop Town. A few weeks into the season, manager Martin McIntosh was sacked; he took over at another Premier Division club, Buxton, in February 2012, and promptly took Roberts on loan. Roberts played his part in Buxton avoiding relegation, and returned to Worksop where he appeared on the winning side in the Sheffield Senior Cup final. For 2012–13, Worksop's manager felt he had better defensive lineups without Roberts, who was keen to play regularly, so he was allowed to rejoin Buxton on loan for the season. He took his statistics to 5 goals from 65 appearances in all competitions, and was named Buxton's Player of the Year and Players' Player of the Year, as well as gaining selection for the Northern Premier League Premier DivisionTeam of the Year. Roberts signed a permanent contract with Buxton in June 2013; although his Worksop contract had expired, a development fee was payable because of his age. A month later, Roberts moved to FC Halifax Town, newly promoted to the Conference Premier. Roberts spent two seasons with the Shaymen before signing for League One club Barnsley in May 2015. He made his Football League debut on 8 August 2015 in a 3–1 defeat against Chesterfield. Roberts became Barnsley captain on 28 January 2017, after Conor Hourihane's transfer to Aston Villa. Roberts signed a five-year contract with Championship club Birmingham City on 1 July 2017; the fee was undisclosed. He went straight into the starting eleven for the opening fixture of the 2017–18 season, which Birmingham lost 1–0 away to Ipswich Town, and was a regular in the side until February 2018. When Garry Monk took over as manager, he "had just one day to pick the back pairing" and chose Michael Morrison and Harlee Dean. Monk said later that Roberts had been unfortunate, but that the consistency of the Morrison–Dean partnership meant they kept their places. With Dean suspended for two of the last three matches of the season, Roberts came into the side to face Sheffield United and scored a first-half equaliser in a win that made a major contribution to their avoiding relegation. Morrison and Dean remained the pairing of choice, but when Morrison injured a hamstring in December 2018, Roberts was unable to take his chance. Nine minutes into his first league start of the season, he suffered a calf injury that left him unfit for selection until after the March 2019 international break. He came back into the side for the last two matches of the season with the team secure from relegation, partnering Wes Harding at centre-back after Dean had surgery on a groin problem and Morrison picked up an ankle injury.
