The 22-year-old full-back, who made eight appearances on loan for the U’s last season, has moved permanently following his release from Rotherham and agreed a two-year contract.
Brindley told the club website: “I’m looking forward to next season and being a part of this club. I enjoyed my loan spell and hope to have the same sorts of results over the next couple of years.
“At the end of last season, it was a tough battle for the team, but a good experience for me, but now I start afresh and hopefully push on.
“The lads are a really good bunch, and the staff and management have helped me massively. Being a part of this team is something I’m looking forward to.
“I am ready for regular first-team football. It didn’t quite work out for me at Rotherham, but I am grateful to have the chance to develop and reach my potential.”