The area is well served by public transport. The R68 bus stops directly outside the pool. It runs between Hampton Court Station (connecting to SW Trains) and Richmond Station (with connections to both overground trains and the District Line of the London Underground. From there it passes through Twickenham and Teddington, in both cases not far from the stations.
There are also buses from Staines which stop close by.
You can check further details on the Transport for London
or Red Planet Travel