Well there are plenty of folks with money that are using helicopters to pick them up from Farnborough, Luton and other points to drop them in London and vice versa. Current options are pretty much limited to Battersea.
In my experience most of the passengers are heading for the West End and therefore far West of London Bridge. That said LCY has never and will never open for rotary wing traffic so there is some demand in the Docklands area. I'm sure some heli operators will be abale to give some input on this for you. London is a very poorly served City with regard helipads for general use.
You are correct about the noise lobby against helicopters.
I do seem to remember some sort of project like this before.