The game as given by Kytrad is very similar to a version of "Watch and Chain" found in Shipley in Sussex, 1892 (Opies, "The Singing Game," p. 68-69).
The tune is London Bridge. The "children formed themselves into two lines and, alternating with two or three children who called themselves robbers and swaggered between the lines, sang as follows:

"Hark! at the robbers going through;
Through, through, through;
Through, through, through;
Hark! at the robbers going through;
My fair lady.

Robbers:
What have the robbers done to you;
You, you, you; you, you, you?
What have the robbers done to you,
My fair lady?

Lines:
Stole my gold watch and chain;
Chain, chain, chain;
Chain, chain, chain;
Stole my gold watch and chain,
My fair lady.

Robbers:
How many pounds will set us free;
Free, free, free; free, free, free?
How many pounds will set us free,
My fair lady?

Lines:
A hundred pounds will set you free;
Free, free, free; free, free, free;
A hundred pounds will set you free,
My fair lady.

Robbers:
We have not a hundred pounds,
Pounds, pounds, pounds;
Pounds, pounds, pounds;
We have not a hundred pounds,
My fair lady.

Lines:
Then to prison you must go;
Go, go, go; go, go, go;
Then to prison you must go,
My fair lady.

Robbers:
To prison we will not go;
Go, go, go; go, go, go;
To prison we will not go,
My fair lady.

"After uttering their defiance, the robbers rushed away with the rest of the children after them, and whoever was caught was put in an imaginary prison."

It seems amazing to me that these games kept many of the same words and actions on both sides of the Atlantic, and in Australia-New Zealand. It came to America quite early; Mother Goose's Melodies was circulated there.

Newell says that the game was mentioned by Rabelais as "Fallen Bridge" in 1533.

Newell goes on to discuss the Middle Ages belief that the soul, separated from the body, had to cross a dangerous bridge and "undergo a weighing in the balance." The fall of the bridge was ascribed to the malice of the Devil. Whether any of this is really related to the "London Bridge" game or is just speculation, I don't know.
W. W. Newell, 1883 (Dover reprint), "Games and Songs of American Children," pp. 204-211.