I'm not sure about the North of the river bit but from first hand experience from Putney Bridge: lower Richmond Rd / Queens Ride / Priory lane (stay off the cycling lane) / Roehampton gate / Richmond park (stay off deers, bunnies and triathletes) / Kingston gate / Queens rd / A wee bit of K. Hill / Old London Rd / Penrhyn Rd via wherever you fancy (might be quicker to go via Villiers Rd, it was for me).
I'm not sure about the North of the river bit but from first hand experience from Putney Bridge: lower Richmond Rd / Queens Ride / Priory lane (stay off the cycling lane) / Roehampton gate / Richmond park (stay off deers, bunnies and triathletes) / Kingston gate / Queens rd / A wee bit of K. Hill / Old London Rd / Penrhyn Rd via wherever you fancy (might be quicker to go via Villiers Rd, it was for me).