Financial details were not disclosed, but British media said Forest have paid £45 million for the 23-year-old. The deal includes a further £5 million in add-ons.
Forest's previous record signing was Omari Hutchinson, who they signed from Ipswich Town for £37.5 million last year.
"I really can't wait to get started, this is a massive club with a big history," Delap said in a statement.
"I spoke to the manager (Oliver Glasner) and I like his way, I think he can be the one to really help me develop and make that next step and hopefully I can bring goals here."
Delap, son of former Stoke City midfielder Rory, came through Manchester City's academy and joined Ipswich in 2024, scoring 12 goals in 37 appearances in the 2024/25 season.
He moved to Chelsea in 2025 and was part of their Club World Cup-winning squad but failed to nail down a spot in the starting lineup as he struggled with injuries and poor form, scoring three goals in 47 appearances across all competitions.
Forest, who finished 16th in the Premier League last season, travel to Liverpool on Saturday.