This is what the Opies say about the game in "The Singing Game," pp. 379-382.

"This is the most eccentric of the eccentric ring games: it has changed in shape at least four times in the past hundred and sixty years.---
"In Mrs. Mackarness's "Young Lady Book," 1876, the 'Muffin Man,' though still played with question and answer round a seated ring, there is no mention of forfeits. And by the time Alice Gomme was sent the game, c. 1894, it was played in a standing circle, with a blindfolded child in the centre who had to touch or catch one of those in the ring and guess who they were (sometimes by asking questions and guessing by the person's voice)."
"The latest but probably not the last shape of the game is a dancing ring, in which... 'One boy or girl stands in the middle and begins by going to someone, dancing in front of them, and singing "Do you know the muffin man...who lives in Drury Lane?" and that person answers "Yes, I know the Muffin man" and they join hands and dance round, but this time...they sing "Two of us know the Muffin Man."

Forfeits Game: first person chants 'to some sing-song tune' "Do you know.. Drury Lane? Next person replies, 'Yes I know etc.' Then both exclaim "Then two of us know...etc. and so on around the ring with more and more people knowing the muffin man. "Merrie Games in Rhyme," E. M. Plunket, 1886.
Often, in the U. S. and The Netherlands, the chooser dances round with a kicking motion of his legs.
First mention of muffin man in the OED is 1810.