Wikipedia page on the London Stones went on to mention two that are not on the Thames at all. For as well as buying rights to that river the City of London also seems to have acquired rights to the River Medway too.
For completeness I decided to add Upnor to the list and above you can see not one but two London Stones, the old and the new, conveniently co-located. The two English Heritage records can be found here and here.
The old one is better preserved than the others so its possible to clearly see the shield of the City of London plus the date 1204 on the front (below) and on the back (above) the words "God Preserve the City of London".
Upnor is very pretty, with a cobbled street and two old pubs:
This part of the Medway has strong maritime connections, with nearby Upnor Castle (which failed to fight off the famous Dutch raid in 1667) and across the river the Chatham Docks:
Just upriver from Upnor is Strood, one end of the Medway and Thames Canal, which was to be my final stone of this particular project - even though it wasn't a London Stone.