Defender Finley Welch made his debut for Norwich City's Under-18s in a 5-0 defeat to Fulham on 27th March 2021. At the time he was part of Norwich's Under-16s. Finley has played for England at Under-15 level. He is from Upminster.

On 16th July 2021 it was confirmed he was awarded a scholarship deal with Norwich.

An article for My Football Writer described his "most influential strength being his ball-playing abilities. Across last season, he played up to 20 games, with a large proportion of them being the lone pivot. Welch has a high footballing IQ, as he’s a player who can receive the ball from the defenders and then pass progressively to the thirds."

On 7th December 2022 he signed his first professional contract with Norwich. He said "I'm lost for words really. It's a big moment for me and my family, a massive thanks to them. It's always been my dream to be a professional footballer, since I was a young boy, and now that it's happened, it's just surreal."

Finley was on the bench for Norwich for their Carabao Cup clash at Bristol City on 29th August 2023.

On 23rd October 2023 he signed a new contract keeping him at Carrow Road until 2026, with an option for a further year. Signing his new deal, Welch said: "I'm so happy. I'm so grateful for the opportunity to sign the second pro contract. I owe a massive thank you to the staff, players, and to my family and friends for helping me get to this point. I can't wait to continue my journey here and continue with the pathway."

Wekch made his Norwich debut on 6th March 2024, coming on as an 87th minute substitute for Ashley Barnes in the 3-1 defeat at Middlesbrough.

On 17th October 2024 he joined National League side Boston United, managed by Ian Culverhouse, on a month's loan.