The history of London Bridge was included in a half-hour programme on TV (shown only in the London region, I think) about the building of London just last night, so some of the info is fresh in my mind.
The bridge in question was the version completed in 1209 (replacing an earlier one which was destroyed by fire in the 1130’s). The rights to collect tolls - and the responsibility for the bridge’s repair & maintenance - were granted by King Henry III to his wife, Eleanor, the “fair lady” of the nursery rhyme. But she was a bit remiss in her responsibilities and the bridge fell into disrepair; hence the bridge was actually “falling down”. The tolls and the repair & maintenance were taken away from Eleanor and given to, I think, the Corporation of London.
It wasn’t a bridge as you’d recognise today: all along its length were houses and shops, with a fairly narrow passageway running through them. These were removed in the 1760’s and the bridge survived until it was decided to replace it with the version that was completed in 1831 and which now stands rebuilt at Lake Havasu.