I did a web search for more children's clapping and singing games, and found the Library of Congress site which has field recordings of folks singing the songs, here.

Their recording of "Mary Mack, Dressed in Black" was sung by an adult prisoner in 1939, so if the song goes back to his childhood, that means it goes back to at least the turn of the century, or thereabouts, I figure.... though it sounds quite different from the version I know -- not in the clapping rhythm I'm familiar with.