The 21-year-old, who had been linked with a host of Championship sides this summer, scored 13 goals in 51 appearances for Walsall last season.
"We are delighted to have Jamie on board. He was very impressive for Walsall last season and has good pedigree," Davies told the club's official website.
"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 is expected to begin pre-season training with Forest this week.