Battersea Power Station will have its own Zone 1 Underground station. Until then, the best options are:
For Circus West Village (via Grosvenor Arch)
Sloane Square (Zone 1, Circle and District lines). A 12 minute walk via Ranelagh Gardens and Chelsea Bridge or a five minute bus ride on routes 452 and 137.
Victoria (Zone 1, Circle, District and Victoria lines). A 20 minute walk via Buckingham Palace Road, Ebury Bridge Road and Chelsea Bridge or a ten minute bus ride on route 44. Victoria is also a major railway terminus and bus station with a dedicated express service connecting to Gatwick Airport.
For the south and east of the Battersea Power Station site
Vauxhall (Zones 1 and 2, Victoria line). A 20 minute walk via Nine Elms Lane or a five minute bus ride on routes 156, 344 and 436.
The nearest bus stop to the entrance at Pump House Lane is 'Sleaford Street'. The following buses serve this stop: 156, 344, 436. From here, cross the road to the entrance at Pump House Lane, and walk along the pedestrian route to Circus West Road (10 mins).
The Northern Line Extension is due to open Autumn 2021.