Define "London". In inner/central London, where cycling infrastructure is being built and where the Congestion Charge is in place, the cycle commute share is significantly higher than 5%. (Back in 2011, for example, it was 15.4% in Hackney: it will be bigger than that now.) In the outer suburbs, where there's no infrastructure and no Congestion Charge, it isn't above 5%. That should tell you something.

Meanwhile, you talk sneeringly about "invest[ing] billions into poorly designed public transport". Public transport is, of course, something London does very well. Car commuting across the whole of London, inner and outer, is around 25%. Public transport commuting is around 50%. (Source: LTDS. This covers London residents only, so is probably an underestimate given the vast numbers of people from outside London who commute into the city by rail.)