There are too many people is the problem, not that there are not enough houses.
The numbers of people are not always the problem, it's the number of people per dwelling which is the problem.
Not that long ago 4 and 5 bedroom houses were pretty much the preserve of those with families, now (having seen those families grow up) it is not uncommon for there to be one or two people in large "family" homes.
Add to that the percentage of people who are not married or living as part of a family (either through choice and/or circumstances) which means that the number of people per house has dropped significantly.
Combining these two factors has lead to more houses needed for a given size of population (say per 100,000 people), meaning that even if our population was falling we would probably still need to be building a lot of houses.
I would argue that most people who have come from overseas to live and work in the UK tend live with more people per property than those who come from the UK.
Of those who are born overseas and are now "living over here" there are quite a few who were born overseas to british parents who moved back to the UK whilst their children were young (i.e. one big group are children of our armed forces) who our immigration system could not (and rightly so) stop from living here.
The numbers of immigrants who we could stop if we were a LOT more strict about letting people in is in is relatively low compared to the overall population. However, even if we did that it wouldn't solve our housing problems in fact (bearing in mind what eastern europeans did to the cost of undertaking building works) there is an argument that it would make it worse.
Likewise with the native population aging (and quite significantly, given that of the population as a whole there are now more people aged over 80 than there are aged under 18) there would be less people to undertake the jobs that need doing, and as such we need our population to be growing to enable our country to keep functioning.
One of the problems with the planning system is that a lot of objectors are based on 1980's mindsets to development (where all the local area got were the new houses and all the problems that caused to local services), whilst now with the levels of funding that developers have to provide (such as to transport, education, social housing, public open space, etc.) the balance as to whether there is a benefit or not to having development is much closer than it used to be.
That's before you take into account that if a borough has 30,000 homes a lot of it's costs are the same as if it has 25,000 homes or 35,000 homes and so economies of scale come into play, meaning that everyones council tax can go a bit further.
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There's something like 500,000 empty properties in the UK.
OK, so some are justifiably so, but many were bought up for redevelopments that never happened, some (especially in London) are held empty as investments so as to deliberately exploit and exacerbate the property bubble, some are lost in not knowing who owns them (eg. investment companies that went bust, overseas buyers), some are stuck in probate. Councils have the power to compulsorily purchase houses and bring them back into use - but this government has leant on them hard not to do it. After all, the absentee owner's right to let his house rot and fall down should trump the poor guy's right to a roof over his head, shouldn't it?
500,000 sounds a lot (current stats have it at getting on for 650,000), until you realise that there are 25 million houses which means that about 2% of all houses in the UK are empty. By the time you allow for some being empty between tenants (sometimes just for a matter of a few months whilst works to improve the properties are carried out) and all the other "justifiable reasons given I would be surprised if there were very many available to actually be useful in housing many more people.
If you look at the numbers involved which have been empty for more than 6 months and you are down to about 200,000 which although would help it would probably only make up one years worth of shortfall if they were all made available. As such, it is not surprising that the authorities aren't focusing too many resources in this area.