UK has massively high house prices - at least in "desirable" areas, and some not so desirable ones where it's possible to commute to work in a desirable area.
That doesn't necessarily reflect a shortage, it also reflects availability of money - and therefore the market bearing higher prices. It also reflects the unevenness of the UK economy: the expensive and unaffordable housing is mostly in and around London (where ~all the economic activity is), and in the most scenic of rural areas which is retirees, second-home owners and AirBnB investors. There's plenty of the UK which has affordable housing, but it's too far from anywhere with decent jobs.
The boom in house prices began when it became the norm for educated women to have lifelong progessional careers. All of a sudden there was a huge amount more discretionary income in the educated class. If you consider that, circa 1970, perhaps 20-30% of one male earner's income went on housing and the rest on life's essentials, all of a sudden the potential amount of cash to spend on housing goes from 30% of one income to 130%, a more than 4x increase. It took a while to filter through, sure, but in the end, it has - to the point where most families are now obligated to have two full-time earners.
Housing is a positional good (think about an auction where demand of the most desirable items will always exceed supply), the prices bear almost no relation to rational economic utility and every relation to how much cost people are able to bear. Which is one reason they're so responsive to interest rates i.e. debt affordability.
A 1.49 birth rate and 685k net migration isn't a desirable situation for any country, but the other issue here is the health and productivity of our own population, we're using migrants to prop it up and provide much of the labour needed by the NHS, childcare and elderly care. Our government is massively anti-immigration, and yet immigration remains high, the country can't and won't go cold-turkey on that: it's not practical to do so without further increasing the retirement age, cutting pensions and increasing "sin taxes" to keep a greater proportion of the population healthy enough to work til 70. And that'd be even less popular, politically, than mass immigration.
The reforms the UK needs are to get the economy functioning better in the regions, and to make it much easier to build higher density housing close to where the jobs are. The current system where they can't build apartment blocks on train-station car parks because a bunch of pensioners complain that it spoils their view is massively counterproductive.