It is a misperception that blue-green infrastructure (BGI) for surface water management will not work in London because of the London Clay. This is to misunderstand the drift geology of London (which is reflected in the soils) and also the functioning of BGI. Although London Clay overlies the Chalk in the centre of the London basin, it is itself overlain on most places by sands and gravels from subsequent river and marine deposits and eroded subsequently into a series of river terraces. Click London geology for British Geological survey map (then "City of London" then "Geology Key").
In cross section it is a sandwich of chalk overlain by clay overlain by sand and gravel. Rainwater infiltrates into the chalk hills of the Chilterns and North Downs and flows down under London – a classic artesian basin. The second misunderstanding is that BGI is not merely storage of rainwater, it also works by slowing the rate of runoff, thus decreasing the height of the peak and spreading the duration of a hydrograph. Some water will infiltrate, some with evapotranspire from the vegetation and some will trickle out over a period of hours. The guiding principle when designing BGI should be the soil and the plants.
Portland, Oregon is an acknowledged leader in the field of BGI (or GI in the US). It too has a complex pattern of drift geology. Even where there is clay at the surface resulting in soils with nearly zero infiltration capacity, Portland has shown that BGI performs successfully coping with far worse storm conditions than London. They have found that the infiltration capacity increased with time since establishment.
What would be the impact of infiltrating far more water rather than allowing it to run-off directly into the sewer systems? The groundwater table is rising beneath London because old boreholes are no longer used. This rising water threatens deeper basements and tunnels and so Thames Water pumps groundwater to control it; they pump it into the sewers! This could instead be seen as a resource, e.g. instead of the desalination plant, as it has little chemical contamination and any faecal contamination from water leaking out of sewers (exfiltration) would be treatable. If sewers are poorly maintained, there may also be water leaking in (infiltration) when soils are wet.
The myth of London Clay being a barrier needs to change into it just being a design issue. Philadelphia's response to their soil conditions has resulted in devices with about 50% direct infiltration and 50% storage with slow release (attenuation). Something similar may be possible for London but further study is required. Tackling leakage (exfiltration and infiltration) would benefit how the sewer system performs and its interaction with the groundwater system.