Campbell stars in the Carole King Musical at the St. James Theatre.
The hit Broadway musical opens in London in February 2015.
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.
Goffin's songwriting partnership with King is depicted in the current Broadway musical Beautiful.
Jerry Seinfeld, Bernadette Peters, and Clive Davis were also on hand to cheer for Jessie Mueller and the cast of the new musical.
The actor-singer is playing 54 Below on Sundays before the Broadway debut of Beautiful — The Carole King Musical, in which he stars.