The 18-year-old, who has been training with Posh in recent weeks, has put pen to paper on a three-year deal at London Road.
Manager Darren Ferguson told the club's official website: "Charlie has got a lot of potential and we look forward to working with him and trying to develop that potential.
"He is a big lad, at 6ft 4in, and is someone we have been looking at in training."
Posh goalkeeping coach Jimmy Walker has also been impressed by the youngster.
He said: "The lad has got enormous potential and we hope he realises it with us. He has never had any full-time coaching, so we feel that working with him on a day to day basis will help him.
"He is one for the future. We have some excellent goalkeepers at the club."