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.
Check out a full gallery of Donnelly's live-drawn cartoons from Sunday night's ceremony.
Clint Eastwood, Kevin Bacon, Will Ferrell, and more have also made the Tony lineup.
Larsen plays Weil on Broadway in Beautiful — The Carole King Musical. Both have written work for the stage, but only one of them considers herself a "writer."