And here is another few references from "Games and Songs of American Children" collected and compared by William Wells Newell first published 1883 - this edition Dover 1963 reproduction of the 1903 second edition.

There are a number of references,

Page 206 - 211.

This is not a hard book to find, and not at all expensive. The one I am reading from is £5.00 ($9.50). It is sold, sorry. But plenty on the book databases like ABE has copies as low as $1.00 !!!!!

There is A "Charlestown Bridge" - Chorus - Dance o'er my Lady Lee,
B London Bridge is broken down - Chorus - Dance o'er my Lady Lee,
C London Bridge is falling down - Chorus - My fair lady
D With tune Same
E from Savannah of Irish origin - Same.

Dave