There's a direct train Richmond to Wimbledon once an hour or you can change at Clapham Junction to get a Wimbledon train, or take the bus from Wimbledon to Putney and get any number of trains from there. A thirty minute commute in secondary really isn't very much.
Alternatively, lots of people from Richmond go to Putney High as it's so easy to get to on the train.
Am I right in thinking WHS does not have any private coach routes? Cannot find any mention of it on the web site. We live in Richmond so am thinking is not feasible if public transport the only option.