The 19-year-old, who was released from the Gunners without making a first-team appearance, links up with the Shakers after previously enjoying a trial at Bristol City earlier in the summer.
Blackwell said: "We've signed him as he has some great potential down the line.
"He was at one of (former goalkeeping coach) Fred Barber's soccer camps and he has been very impressed with him, we haven't got a young goalkeeper and Reice fits the bill."
Charles-Cook put pen to paper ahead of the Shakers' 3-1 friendly defeat to Sheffield United on Saturday.
Striker Shaun Harrad, who spent last season on loan at Cheltenham, opened the scoring in the first half before the Blades hit back after the break.