London Bridge is falling down
Falling down, falling down
London Bridge is falling down
My fair lady

Okay, my view on this:

1) Okay, the bridge used to be built of wood, and during one winter, the river of thames froze.
A fire started, a bit away from the bridge. The people were like, it's a slow moving fire, we can make it across the river. An ember from the fire lit up one of the spires of the bridge, people started getting worried, but they kept moving. And then an ember from the burning spire, lit the other spire on fire. 

The river was frozen, so they can't jump (If the river was not frozen, you can jump. You won't die, but it would hurt on impact)

2) My fair lady, The spires were rented out to people that can afford it, and a Lord and his wife (The Lady) was living in the second one. The Lord was out doing business, and the Lady was in the spire when it started burning. So she probably died and that's probably why it says My fair lady, out of nowhere