He originally joined Linnets from Welsh side Flint Town in the summer of 2020 and went on to make 50 starts for the club, the last of which came in the FA Trophy earlier this week.
He also played an instrumental role in their journey to the play-offs last season.
Welsh, who can operate at centre-back or full-back, has now decided to move onto pastures new.
He was previously on professional terms at Tranmere Rovers, where he first met recently-appointed Bucks boss Ste McNulty.
Welsh could make his debut in the Pitching In NPL West Division against Workington this weekend.
"I know Bootle are always a very tough side to play against," he said.
"They beat us last year at the Berry Street Garage Stadium and it was a very tough game at Runcorn. They play similar styles which suit my game.
"I can’t wait to get started. It’s a new challenge for myself. I'm itching to play in front of the fans and show the players and manager what I can bring to the team."
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