It's been a long time since I thought about them. Since I never had children, I never got the chance to share any. Though not a nursery rhyme or song, one thing I did do to my husband not long ago was common for parents to do to little kids or even babies. They'd grab each toe (one at a time) and say the following:

Big toe: "This little piggy went to the market."

Second toe: "This little piggy stayed home."

Middle toe: "This little piggy had roast beef."

Second to last toe: "This little piggy had none."

Little toe: "This little piggy went 'Wee, wee, wee....all the way home!" [And then usually my mother would run her fingers (like running) all the way from my little toe to my face.

My Czech husband said they do something somewhat similar in Czech Republic.

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A really old song that I always liked surely originated in England, or somewhere in the UK. It is always sung at Christmas time. It goes:

"Christmas is coming, the geese are getting fat. Please put a penny in the old man's hat. If you have no penny, a half-penny will do. If you have no half-penny, God Bless You!"