Barely remember it, but I remember grade-school girls singing it.
This is from Opie and Opie, "The Singing Game," pp. 61-67, with music.

(Explanation by nine-year-olds) "You make a bridge with two people, and all the other people line up, and you sing "London Bridge is falling down" and they all go round under the bridge...And when you get to "My fair lady" you catch somebody. And then when you're singing "Sticks and stones will wear away" you swing them about backwards and forwards and round and round - any way you like- as if you've got them in a sieve. And then you take them away and they whisper what they'll have, 'sticks or stones,' and they go to whatever side of the bridge they've chosen. Then you sing "London Bridge" again, but first you whisper what the next verse is going to be, and which person is going to be which thing. The bridge never is mended but it doesn't matter. You have a tug of war at the end."

The Opies give examples from the 18th c and cite its existence in the 17th c. The song first appeared in "Tommy Thumb's Pretty Song Book," ca. 1744.

London Bridge is broken down,
Dance over my Lady Lee,
London Bridge is broken down,
With a gay lady.
How shall we build it up again...
Build it up with gravel and stone...
Gravel and stone will wash away...
Build it up with iron and steel...
Iron and steel will bend and bow...
Build it up with silver and gold...
Silver and gold will be stolen away...
Then we'll get a man to watch.

The song became a ring dance by the 1820s.

Lots of verses and variants. Surprising that there doesn't seem to be a thread on the song and games.