Well, there are version dating back to the middle of the 18th century, but the version that you most often hear today comes from the 1950's.

The actual meaning of the rhyme is unknown, the version that I most often hear is that it was based on a Viking sack of London in the 1000's, with the subsequent destruction of London Bridge. This in turn comes from a Norse saga that I forget the name of (sorry!). However, there isn't really enough evidence to ever prove that this attack happened.