So the article claims that the housing construction boom that ended at the second WW led to a massive stock of housing, which has since declined due to planning issues, leading to a shortage of housing and therefore a rise in prices.

This does not hold up statistically.

1939: 11.3 million dwellings in the UK, population 47.9 million people = 4.2 people per dwelling

today: 27 million dwellings in the UK, population 67.7 million people = 2.5 people per dwelling

A thought occurred: maybe there are now fewer people per household. This however does not hold up to scrutiny either. The number of UK households rose 19% in 27 years, a rate comparable to the increase in housing stock.

If anything there is more housing than ever in the UK relative to population size. If prices have risen, it's not because of a lack of housing.