The 21-year-old has so far made 38 first-team appearances for Forest, scoring five goals, and it is hoped the relevant paperwork should all be completed in time for Saturday's clash against Rochdale.
Yates told the club's official website: "He is a talented young player who we have known for a number of years. He will certainly add something different to the squad and can contribute to the team in a big way."
Richardson, meanwhile, has parted ways with the Robins having been released from his non-contract registration after the club opted to operate without a goalkeeping coach for the remainder of the season. A new appointment is expected to be made in the summer.
The 40-year-old also regularly appeared on the bench for Cheltenham this season but Yates has other options as back up to Scott Brown, with Will Puddy currently on loan at Oxford City and scholarship player Joe Perry an emerging talent.
Source: Team Talk
Source: Team Talk