My children came back from church one afternoon this past year playing a game that went something like this:
Down by the banks of the hanky panky
Bullfrog jumped from bank to bank
Hip hop soda pop, hey Mr. William and he went kerplop!
They sat in a circle, criss cross applesauce and had their hands facing upwards on their knees, right hand resting in the left hand of the person on their right. As they sang the song, one person would slap their right hand towards their left hand, and slap the right hand of the person on their left. The slapping would go around and whoever it landed on at 'kerplop!' would be out.
I was amazed at how quickly they had picked it up and wonder at how these songs have a way of traveling orally from generation to generation. I remember as a child sitting in the back of the school bus in Japan on the way to the American school hand clapping with a friend to 'Miss Mary Mack'.
Charlotte talked about these types of games that she had seen played in England in her time:
Before Puritan innovations made us a staid and circumspect people, English lads and lasses of all ages danced out little dramas on the village green, accompanying themselves with the words and airs of just such rondes as the French children sing to-day. We have a few of them left still––to be heard at Sunday-school treats and other gatherings of the children,––and they are well worth preserving: 'There came three dukes a-riding, a-riding, a-riding': 'Oranges and lemons, say the bells of St. Clement's'; 'Here we come gathering nuts in May'; 'What has my poor prisoner done?' and many more, all set to delightful sing-song airs that little feet trip to merrily, the more so for the pleasant titillation of the words––dukes, nuts, oranges,––who could not go to the tune of such ideas?
The promoters of the kindergarten system have done much to introduce games of this, or rather of a more educational kind; but is it not a fact that the singing games of the kindergarten are apt to be somewhat inane? Also, it is doubtful how far the prettiest plays, learnt at school and from a teacher, will take hold of the children as do the games which have been passed on from hand to hand through an endless chain of children, and are not be found in the print-books at all.
It seems the games we know are more likely to have come out of the latter description here. But as you are anxiously preparing for academics with your little ones, perhaps teaching your children some of these games would be another way to enrich their year.
Here is a link to a book called "A Dictionary of British Folklore" http://books.google.com/books?id=e4ohsoURGnEC&lpg=PA338&ots=RxuavsWfZY&dq=Burne that has a large list of nursery rhymes/games that have been observed. (Warning: There are kissing games and lyrics mentioning chopping off of heads and such so obviously, use discretion as needed.)
Here is a video about the rhyme 'Oranges and Lemons' which Mason mentions in her quote above: