Downriver from Putney Bridge is Fulham Railway Bridge. It dates from 1889 and has a footpath, accessed by three or four flights of steep metal steps at either end. Not a problem for featherweight fixies, but no easy task on a solid tourer with panniers full of shopping.
Past the bridge, on the north side it's wiggly roads up to Wandsworth Bridge. On the south side it's cycle paths towards and through Wandsworth Park, and then wiggly roads to Wandsworth Bridge.