Mueller's Tony-nominated costars Jarrod Spector and Anika Larsen have also extended their contracts.
Campbell will replace Jake Epstein in the role of Gerry Goffin.
The new musical will open in the West End at London's Aldwych Theatre in 2015.
Jessie Mueller, Douglas McGrath, and Steve Sidwell eulogize the classic song.
The Harry Potter star will reprise his performance in Martin McDonagh's The Cripple of Inishmaan in April.