https://sf.curbed.com/2016/10/31/13481254/micropad-tour-patr...
The problem is that the transport network is at capacity: road, bus, train, tube - everything is packed at peak time, the main roads are a continuous stream of busses, there are 3 train in 20 minutes, a dlr every 3 minutes, ... If you double the density, you need to double the transport links too.
Then there are the nurseries/schools/hospital. My GP has literally thousands of people registered despite a 3 street catchment area. It take at least a month to get a GP appointment.
What's the point of living in a cheapass flat if you need to pay 2000 GBP for nursery or similar for private school, you need to go to private GP when you get a cold and you are still miserable in the public transport and ambulance do not get to you faster than if you were in the middle of the country side. London is at capacity for people working in the City/WestEnd/CanaryWharf. You need to expand it outward (eg: Crossrail ) and start encouraging business centers off the convergence commute route.
BTW I don't think the cost of building is really a problem. The new build quality is appalling, a luxury unit cost little more than 50K to build and will be sold off plan for close to 1 million, years before the builder gets the council approval.
Could not transportable housing be sited on short lease plots, with the aim of being a rudimentary means to live in a city but without having permanent planning or zoning decisions implemented?
Certainly if I was younger, I would be interested in prefab housing for say 5 years lease on the site, and not expecting any permanent conversion or to be a squatting occupier, if I faced the London market at present again. Location is so important, and I pass so many empty lots even in the city's central Whitechapel area as I walk into the City centre. I would probably have been thrilled to meet my needs at twenty, this way.