Route 21 starts at Greenwich/Deptford and goes through the Waterlink Way, and eventually finds the Cuckoo Trail and then ultimately the south coast at Eastbourne.
The London Cycle network has got a quiet route which goes from south-east London and through various parks : Ruskin, Brockwell Park, Clapham Common, Wimbledon and then into Richmond. Just follow the blue signs with the white bicycle.
There is also quiet routes which go up through Rotherhithe and through the ecological park there.
If I were you I would not go anywhere near the Greenwich Foot tunnel, which although it claims it has re-opened it is in varying degrees of lifts operating/not operating/stairs operating/not operating, its just a nightmare at the mo. Also the various canals around the Olympics are still in a 'special' way too.