It takes two parties to increase the price of a housing unit. If rents are up 20% that means people are willing to pay 20% more.
I don't know about the UK, but here in California, prices have skyrocketed, but so has the price of construction.
There are two reasons that has happened:
1) Shortage of construction workers. The ones that are here are either (well paid) first generation immigrants, make more than Silicon Valley windfall money because they speak fluent English and are competent, or are completely incompetent. We don't have enough of the first group, so be prepared to pay 2-3x the cost of materials to get anything done (with the second group acting as middle men, and the third group taking your money, only to have you pay someone else to fix it later).
2) Policies that discourage the building of new rental units:
2a) We have a vacant house on our property. A non-structural remodel for it would cost > $50K to get through permitting, and we're looking at >> $300 sq/ft for the actual remodel. If we did all that, we could either pay a special tax on vacation homes owned by individuals in unincorporated areas that the townies just passed (to help the housing crisis by somehow freeing up housing units in town, where the tax does not apply), or we could rent it out. If we rented it out to a problem tenant, we could literally never get rid of them (unless they decided to not pay rent, but even then, it's 5+ years of court battles). So the house stays vacant.
2b) We just built a house. It took almost a year to clear permitting, and $100K's of wasted nonsense work. Many developable plots in this area are purchased, planning bankrupts the new owners, and then they're sold to the next saps. According to the neighbors, getting permits to build a house around here in under 3 years is unheard of. The result? We have a house, but we are way, way, under water in terms of money put in vs. current valuation. At the lower valuations, the houses around here are not "affordable" by any means. However, if you look at what it would have cost us to develop this land anywhere else in the country, we would have paid way under market value.
Problem (1) could be fixed by encouraging contractors from out of the state to fly in to work here. Apparently, the state has erected licensing barriers to make this hard. I think a lot of money ($1B) is waiting to be taken off the table via arbitrage.
Problem (2) is consistently worsened by voters that think that capping housing costs, "protecting tenants" and other things that further constrict housing supply will somehow lower prices.
The easiest way out of this problem is (1) allow out-of-state firms to build housing and (2) fast-track all new housing permits, including financial liability if the planning commission creates unnecessary delays, unnecessary work, or approves / passes inspection on substandard work.
They should also replace the state wide mandate to reduce commuter miles (which is basically a mandate to increase congestion by tearing out roads) with a mandate to reduce the total carbon emissions per capita spent on transit (which would be a mandate to invest in public transit, bike lanes, and in reducing congestion).
None of those things are politically tenable, so I guess the millennials will just live in RVs or 4-to-a-bedroom until the voting population turns over.