I don't think it works at all in your case, but having been involved in a small way with social housing in one of the London Boroughs, it's not a bad scheme.
Where it works particularly well is when you have an elderly tenant whose family have all grown-up and they are rattling around in a 3/4 bedroom house - the offer of £50k to move to a 1 bed flat (in the same area - and often with a redecoration grant) is a great deal for both sides, even if it seems grossly unfair to many local taxpayers and people in private rental.
I admit to getting very irritated when people in that situation didn't want to move as they liked the extra bedrooms for their grandchild to visit once a year. When you have seen families with 3 children stuck in a 2 bed flat for years due to severe shortage of larger houses, it seems ludicrous that someone can refuse to move from a house that they don't need and don't own.
Especially when you have 15k people on the housing waiting list and no land or money to build more then it can help. Councils would never be able to compete with private developers for available building land in central London.
The deal is that developers have to set aside 20% of units for "affordable housing" if their development goes over a certain number. In practice, these are generally 'shared ownership' not council properties. They also pay Section 106 money towards local amenities, doctors surgeries, schools etc
A PP noted that some social tenants can no longer use communal areas they could previously. What actually happens is that when a site is redeveloped, social housing is normally in a separate block from the private housing. The reason for this is that council tenants can't afford the often £2k+ annual service charge that is slapped on the private tenants.
It's illegal to make some people pay more than others for service charges ie subsidising the social tenants. So, the private tenants will get the landscaped gardens and carpeted hallways as they are paying through the nose for them.
For the OP, stay put for definite, a 3-bed in N London is pretty much a lottery jackpot, especially if there are health concerns.