I came across this version while looking for something else. It was too good to pass up.

London Bridge

1. "London bridge is falling down, falling down, falling down. London bridge is falling down. My fair lady!"
   
2. "How shall we build it up? etc."
   
3. "Build it up with wood and stone, etc."
   
4. "Wood and stone would wear away, etc."
   
5. "Build it up with iron bars, etc."
   
6. "Iron bars would bend and break, etc."
   
7. "Build it up with silver and gold, etc."
   
8. "Silver and gold'd be stolen away, etc."
   
9. "Set a man to watch at night, etc."
   
10. "Suppose the man should fall asleep, etc."
    
11. "Put a pipe right in his month, etc."
    
12. "Suppose the pipe should fall right out, etc."
    
13. "Stick it in with Spalding's glue, etc."

Source: Vermont Folk-Songs & Ballads, edited by Helen Hartness Flanders & George Brown (1931), page 45

Notes:

SENT TO MRS. FLANDERS by Mrs. R. M. Wilcomb, Springfield, Vermont, March 25, 1931, who wrote Grandmother, in writing it out for my mother, fifty years ago, finished with the verse about "Spalding's glue," and what more, I'm sure I don't know. Mrs. Wilcomb grew up in Lyndon, Vermont. Her grandmother came from Gloucester, Mass.