When I first began flying helicopters, IIRC, there was a barge helipad on the Thames, up near where HMS Belfast is now moored.
Battersea is being improved and is much better than the one pictured in Sweden.
It certainly does need some competition though, the fees are ridiculously high.
The main issue is obtaining planning permission. Unfortunately in UK, helicopters are seen by many as a nuisance and an unnecessary plaything of the rich. The previous Lord Mayor, Red Ken, would apparently have preferred to have banned all of them from the Capital.