The London Bridge:
Obviously, this is not the Medieval London Bridge. In fact, this bridge, completed in 1973, is only the latest of several bridges to bear the name of "London Bridge."
What about the one before this one? That bridge is the "London Bridge" that was dismantled, exported, and reassembled at Lake Havasu in Arizona. But that one isn't the medieval bridge, either. Designed by John Rennie and built by his son , it was completed in 1831 as a replacement for the medieval bridge. The medieval bridge was dismantled utterly after the "New Bridge" was opened, and aside from a few carved objects made as souvenirs from some of the original pilings (and various pieces of stonework that were actually eighteenth-century additions to the original bridge), nothing remains but pictures.
It's impossible to say when the Thames was first bridged at that location. Evidence from Roman times (consisting mainly of Roman coins found on the riverbed during the building of the 1831 bridge) suggests that the Romans might have built a bridge, but it is equally possible that the coins might have come from the sinking of a ferry. Also found were some iron "piling tips" but these might have come from any of several wooden bridges which were built in the last few centuries of the first millennium.
The earliest reference to a bridge is in a Saxon record; it was mentioned as an execution site for a witch during the time when one Aethelwold was bishop of Winchester, which would have been sometime between 963 and 984.