Atletico Madrid considered bringing Fernando Torres back to the club over the summer, but made no formal approach.
Stefan Johansen is feeling settled at Celtic and has no intention of pushing for a move any time soon.
The 21-year-old has signed a two-year deal at the City Ground with the option of a further year.
Forest boss Billy Davies expects Paterson to adapt quickly to his new surroundings as Forest hope to push for promotion this season.
Davies said: "We are delighted to have Jamie on board. He was very impressive for Walsall last season and has good pedigree.
"He's quick, agile and has proven himself as a composed figure in front of goal. He adds another dimension to our attack.
"While Jamie is undoubtedly an exciting prospect, he has already gained good first team experience and that should help him make the step up to the Championship."
Paterson could feature against his old club on July 23, when Forest travel to The Banks's Stadium for a pre-season friendly.