Fletcher, a former contestant who wrote the music and lyrics for the new Paddington musical, joins other special guests, including the cast of the Olivier Award-winning musical parody, Titanique.
The show will feature a new dance relay, a first for the competition, involving the five remaining contestants performing a five-minute routine choreographed by Matt Flint.
The relay will see couples like Amber Davies and Nikita Kuzmin, Balvinder Sopal and Julian Caillon and Karen Carney and Carlos Gu perform routines from various musicals.
Doctor Who star Alex Kingston and professional dancer Johannes Radebe were eliminated in the previous episode, with the next shows airing on Saturday at 6:50 pm and Sunday at 7:45 pm on BBC One and BBC iPlayer.