Not just South Western Railway, but Southern stations too. Vauxhall and West Croydon don't have toilets, for example (in both cases the adjacent bus station does, though).
For many years Balham station had no toilets even though neighbouring Wandsworth Common (where Sutton via Mitcham Junction trains don't call) does still have them. According to the National Rail website Balham does now have an accessible toilet, though (probably a similar situation to Putney).
Most smaller suburban stations in London and other major cities gradually lost their toilets between the 1970s and the 1990s. To be fair, though, you probably wouldn't class stations such as Queenstown Road (Battersea), North Sheen, Hampton Wick or Berrylands as "important" stations.