The myth has reared its ugly head a few times recently, so let's have another go at busting it.
Thankfully, Economics Help has done a good article on this to save me the bother.
To cut a long story short, whether there is a housing “shortage” or not is not a serious question. Most people would prefer to live in a larger home than the one they do, so what? That goes for most things. But assuming that somehow there is an objectively assessed shortage, we would expect rents to have risen dramatically and they have not. They have gone up in line with wages, eating up approx. one third of monthly extra net incomes after tax (I'm surprised it's that low, but there you go). Although extra supply in the last ten or fifteen years was pitiful, it has been just about enough to cope with population increases (assuming two people per new dwelling).
What has risen is the multiple of rental value which people will pay, which by and large is the inverse of interest rates. Interest rates have fallen to more or less zero, adjusted for inflation, so the multiple has gone up from ten to thirty.
(I accept that London is an extreme case, but I've explained that one to death as well and it is not really relevant to the rest of the UK.)
End of discussion.