The "variable location" idea was mooted, but the City Heliport never did move its location - and when the adjacent site on the north bank was developed, the temporary planning permission went with it. The City of London school is on that site today. The City HP was capable of taking everything up to S76s.
Star Aviation had their own facility with two barges each with two decks, and was on the south bank, just west of Southwark bridge prior to the buildings there being developed. Those barges were capable of 206s only as I recall - it predated Star's AS355F2 anyway, and I visited once when 206s occupied three of the four pads.
Various other sites have been tried, including a BEA Helicopters S-55 operating from the South Bank, possibly from Jubilee Gardens - perhaps this was FL's one in the Lambeth area - it's not that far from Lambeth Palace!
Noise trials were also run in the late 80s with a 76 and a barge on the south bank approx 0.5 miles E of Tower Bridge - from memory that was known as "Chamber St Heliport", but nothing came of it.
Another idea had a HP on top of Cannon Street Station - not sure why that failed, but it seemed sensible to try and beat the NIMBYs by putting two noise sources together
Then there is Medina Aviation operating out of Battersea Power Station site, but presumably that holds just as long as the site remains in its current state only - and is, of course, not ideally located for the City customers.
Now, here's an idea.... Given that they are trying to find uses for the Millenium Dome site, and it has good tube connections, why hasn't someone thought of that? Is the south bank location really that bad?
What about an aviation industry lobby to get helicopters allowed at the City Airport?