Going back to the 1950s there was a floating helipad between Lambeth and Westminster bridges, and in the about 20 years ago there was another one at Cannon Street, but it was closed down by the noise/safety lobby.
The BHAB has identified seven suitable sites along the river, and had high hopes of one at Gallions Point, on the south east corner of EGLC, but I believe that has now been built on.
No government (of any stripe) wants to know about this one, simple as that.
The BHAB has lobbied to have EGLC opened to helicopters but looks to be on a loser. The airport was originally given planning permission on the condition that no jets were allowed, and of course they have the 7-degree glideslope for noise abatement. Since that original decision, of course, jets have been allowed in - originally it was only the BAe 146, but in time it became fair game for just about anybody. But what chance of helicopters ever getting in there? Diddly squit, IMHO