I lived in Twickenham for many years. Eel Pie Island to Richmond Bridge is also one I've done more times than I can remember, usually involving stops at multiple pubs along the way.
I'd run between Putney and Twickenham for my marathon training solely because the incredible beauty would make the pain and loneliness so much more bearable.
Also great as a cycling path - Putney to Hampton Court Palace and back!