Hammersmith is currently served by the District, Piccadilly, Circle and Hammersmith and City tube lines. It is a 20 minute walk from the tube station to the school site at 80 Lonsdale Road. NB as you may be aware, Hammersmith Bridge is currently closed for repairs, so buses that previously ran from Hammersmith Tube station to Castlenau or Lonsdale Road are not currently in operation.
Barnes Station is served by frequent trains from Clapham, Waterloo and Richmond. You can take the 33 or 419 bus from Barnes station to St Paul’s, getting off at the Lonsdale Road stop.