The 31-year-old goalkeeper signed a three-year deal at the Championship club after Watford opted to allow him to depart for an undisclosed fee.
"From day one I said it is an honour to work with the manager (Aitor Karanka). I really want to spend as much time as possible working with him," he told the Forest website.
"This club as well, I said from day one that the fans and everyone at the club really appreciate me and my work and showed me true love.
"This, in my opinion, is the most important thing for a player.
"I am so excited to get started again and I am glad to be part of this project again, so I am really excited to be at the training ground."