Children have played this game - two arms make the bridge, everyone else runs under before it falls - since before anyone thought to write it down. A play mentions the broken bridge in 1725; a tiny songbook printed the rhyme in 1744; and the exact verses here were collected from schoolchildren in Berkshire in 1894. The real London Bridge fell down, burned down and wore out many times over the centuries, which is either the song's origin or its best joke.
The song is an imperative machine: , , . Notice how English snaps the little word onto the end - build it up, wash it away - that is the phrasal verb pattern, and this song drills it more gently than any textbook.
Two old words to file: mortar is the wet mix that glues bricks together - English still says "bricks and mortar" to mean buildings - and the in "my fair lady" is the old sense of beautiful, not fair as in a fair game. Nobody knows who the lady was. She has been listening to the bridge fall for three hundred years.