Here are the songbook citations noted in the Traditional Ballad Index:

Randolph 578, "London Bridge is Falling Down" (1 text)

London Bridge is broken down,
Dance over Lady Lee
London Bridge is broken down,
With a gay lady.

Shall we build it up again?...

Mud and clay will wash away...

Iron and stone will stand alone....

Flanders/Brown, p. 45, "London Bridge" (1 text)

London Bridge is falling down, falling down, falling down (13 verses)

Linscott, pp. 34-36, "London Bridge" (1 text, 1 tune)
(my copy of Linscott is missing - it's skinny and gets lost)
SHenry H48h, pp. 11-12, "Broken Bridges" (1 text, 1 tune)

Broken bridges falling down, falling down, falling down...my fair lady. (8 verses)

Chase, p. 189, (no title; part of a section called "Granny London Tells About Old Times") (1 text, 1 tune)
(can't find Chase, either)
Creighton-SNewBrunswick 81, "London Bridge" (1 text, 2 tunes)
(don't have this one)
Opie-Oxford2 306, "London Bridge is broken down" (4 texts)
["broken down (3x) and Lady Lee versions]
Baring-Gould-MotherGoose #641, pp. 254-255, "(London Bridge)"
(got it if you need it)
Fuld-WFM, p. 337+, "London Bridge"
(falling down three times)
ST R578 (Full)

Roud #502 (click here)

RECORDINGS:
Pratt children and friends, "London Bridge" (on Ritchie03)
Pete Seeger, "London Bridge" (on PeteSeeger33, PeteSeegerCD03)
(falling down three times - not Peggy's version)
CROSS-REFERENCES:
cf. "Rock-A-By Ladies" (tune & meter)
SAME TUNE:
Greenberg Shop is Moving South (Greenway-AFP, p. 126 note)