According to the section "Tidal Effects"is website:
http://www.the-river-thames.co.uk/weather.htm
the Thames is essentially freshwater as far downstream as Battersea.
The page also states that above Teddington the River contains fresh water whereas below Teddington the water is saline but the degree of salinity changes with the state of the tide and the volume of fresh water coming downstream, which confirms what Peter said.
I think brackish is probably the best description at London Bridge.