Sorry to put a dampener on this but having tried to crack this one repeatedly over the last couple of decades my conclusion is that there is no straightforward access to central London by light aircraft or helicopter.
None of the peripheral airports have a simple public transport link to central London (i:e walking distance) - all involve a taxi in my experience either to the train/tube or to final destination - and most close at dusk.
As for Battersea - £210 landing fee for an R44 then half that per hour parking plus early surcharge plus late surcharge - and all to end up in a non-central part of London with no public transport links - great! The other heliport at Dagenham I have not been in to but it is also a long way out to the East.
London is an exceptionally poor destination for me to get in to by light aircraft.
Let me know if I am missing something.
For the forseeable future I will be taking the train.
SB