Increases in housing demand (without increases in supply) will almost always cause house price increases by pushing more marginal buyers into the buying market.
Without any increase in supply, more people wanting emergency accommodation or council homes increase wait times. The person on the edge of deciding between council housing or private rental will go for the latter. The new renter will have to try and outbid other renters causing rental rates to go up. Ultimately that nudges the person on the edge of deciding between private rental and buying with a mortgage to enter the buying market and try to outbid other buyers - causing an increase in overall prices.
There's suffering at each stage of this process, where high demand forces people to engage in transactions that stretch them financially. Over time many people will just 'drop out' by staying with parents or going abroad.
The only real solution is to build more houses. In the 1960s the UK built >400k new homes per year [1] - enough to comfortably a million new migrants annually whilst also increasing house choice and affordability for people already resident.
[1] https://twitter.com/bbcquestiontime/status/12041546940519997... (chart is for England, I've extrapolated a peak of 350k for England to justify a claim of >400k for the wider UK)