The Waterlink Way is an 8 mile (12 km) path between Sydenham and the Thames, and it can be walked or cycled. (Large scale map.)
A walk along the Ravensbourne River is also the Waterlink Way until the middle of the River Pool Linear Park where the River Pool meets the Ravensbourne. At this point the Waterlink Way continues along the River Pool to Lower Sydenham. The river continues just beyond Elmers End railway station and is c.3 miles in length – I have not yet walked to the source but here is someone who did.
You may be interested in
The Waterlink Way
The Ravensbourne River – Posts here, here, and here
The River Pool
London’s Lost Rivers