In the early stages of his football career, Watt featured in the academy ranks of Liverpool, Blackburn and Preston North End before signing a professional deal at Wolves in 2017.

Just a few months later, he made his first-team debut for Wolves in the FA Cup. This would be his only appearance for Wolves, with Watt joining Carlisle United on loan in 2020. Although his time in the North of England would be cut short due to the Covid-19 pandemic, Watt still managed to make 14 first-team appearances in his short stint with the Blues.

Upon the return of football in the summer of 2020, Watt joined Bradford City on a two-year deal, who were managed by Stuart McCall at the time.

Spending two years at Valley Parade, he racked up 96 appearances, scoring five goals before joining Salford City in 2022. His impact would be clear to see, as in his first season, he was named in the League Two team of the year. Playing 91 games in total and assisting 24 times, he moved to Burton Albion after two years.

Named captain, Watt made 30 appearances in his first six months with Burton before joining St Johnstone in January 2025. Here, he played nine games, scoring two goals during his time in Perth, one of which came against Motherwell at Fir Park in May.