Not only do I remember FIVE O'CLOCK CLUB, but I have also recently been reading a biography of Alexis Korner, and he certainly was involved, essentially as leader of the house band, which was part of the Blues Incorporated crew. The house band was Korner, Danny Thompson, and Terry Cox, and they not only did spots of their own but also worked with musical guests. Wally Whyton, co-presenter with Muriel Young, was also a musician, having been involved in the skiffle years as a member of the Vipers, who were of some repute.
As to whether Ivor Cutler was involved, I don't know. There's no reference to him in the fairly detailed information about Korner's involvement in the show in the biography. According to the Wikipedia piece on Cutler he was involved in a number of television programmes in the 1960s, though apparently all for the BBC. Not impossible he was on FOCC, but not as yet proven.