I don’t know what the actual situation is in Britain, but here in Ontario the responsibility for public housing is a regional or municipal responsibility. There’s no automatic mechanism for planners in one area to anticipate the need for additional housing, so apartments, townhouses and other subsidized accommodations are informally “swapped” between city, town, and regional governments. Would-be tenants are able to refuse being moved from one municipality to another (if you’re in Oakville, but the offered housing is in Pickering, for example).
I suspect, based on Tim Worstall’s thought experiment here, that the British system does not work quite the same way:
What’s the first thing that rational planner is going to do? Note that there’s a number of people living in London without the means to afford housing in London. And no particular economic reason for living in London either. She’s also going to note that’s there’s great swathes of housing up North which is indeed affordable. And given that there’s no particular economic reason for those in London to be in London why shouldn’t they be on benefits up North in the much cheaper housing?
This will be, after all, greatly to the benefit of society even if a bit tough on the personal liberty side. But then that’s what planning of all these things is about, doing what is best for society, yes?
So you can see the amusement: the Statists, the planners, those who insist that society is more important than the desires of any mere individual, are in something of a bind. The current reforms to the housing market are producing exactly what a rational planner would produce. The poor are sent off to be poor in cheap housing, individual desires be damned.