Raised in Prestwick, South Ayrshire, where he attended Queen Margaret Academy,[3] Watson trained with Kilmarnock from the age of seven and officially joined the club's youth system aged ten.[2] He signed a two-year professional contract on 3 September 2021[4] and made his senior debut the following day as a late substitute in a Scottish Challenge Cup victory over Falkirk, but was not otherwise involved with the first team squad that season as Killie were promoted as winners of the Scottish Championship.
During the following season, he made early appearances in the Scottish League Cup but it was not until March 2023, shortly after turning 18, that he began to feature regularly in the Scottish Premiership,[5] making his league debut under manager Derek McInnes at Ibrox Stadium against Rangers[6] and starting eight of the last ten fixtures; his breakthrough coincided with an upturn in results (with the exception of a home defeat to St Johnstone in which Watson received a first half red card)[7] and the club escaped relegation, finishing in 10th place.[8] He agreed a new two-year contract in May 2023.[9][6]
At the start of the 2023–24 season, Watson continued to feature in the starting lineup and was praised for his performances as Kilmarnock defeated both Rangers (in the Premiership) and Celtic (in the League Cup) by 1–0 scorelines at Rugby Park within the space of a fortnight.[10] In November 2023, he signed a further extension to his deal until 2026 after more impressive performances for the first team.[11] Kilmarnock finished fourth in the Premiership to qualify for the 2024–25 UEFA Europa League and Watson's contributions were recognised with Young Player of the Year awards from both the Writers' Association[12] and the Footballers' Association.[13][14]