snark!?! A damn good word, my grandmother used to chide me with that some 60 or so years ago. I was tempted to look it up- with this meaning it is first found in print in "Railway Children," 1906, but there are other references close to that date.

"Ring around..." appears on the European continent in the 19th c, and in UK as well but perhaps a little later than its European cousins.
In Opie and Opie, "The Singing Game," the many fall down or fall apart games form a group; it may be a barrel full of water splitting (Holland), fruit falling (Europe in general), sitting down to eat potatoes (Spain), dancing around a punch bowl (England), and perhaps the earliest found (1796), from Germany- about three children sitting in an elder bush crying Musch, musch, musch! Sit you down (Rufen alle: musch, musch, musch! Setz euch nieder!). This appeared in England as Hark! they all cry Hush! hush! hush!, Sitty down, sit down.

No reason why there wouldn't be something similar in Africa, Asia, pre-European America.