The 23-year-old, who was part of the triumphant England Under-20s Six Nations side in 2012, has made 22 senior appearances for the club since joining the academy in 2009.
"Will has been an exemplary professional for us," academy director Danny Grewcock told Bath's official website. "He's managed to combine studying for a chemistry degree with being a full-time player, which is no mean feat. We wish him the best of luck with London Welsh."
Welsh will compete in the Championship next season, having failed to win a single game in the top flight this term.