In the UK, housing is mostly provided by the market.* But supply and demand are not the only concepts you need if you want to understand why London house prices are so high, and indeed why they have been rising. You also need to think about expectations and about interest rates.
When you rent housing, you're just consuming the value of the housing: you get a spot to live somewhere. But property also has an investment value: if you buy it, then it will provide you housing services every year until you sell it. A house is very durable, so the value of the services it provides may change large amounts over the period of its life. Its rent—or rental value, if it's not rented out—will go up and down, both because the owner improves or worsens the property, and because the rest of the world changes so that anything located where it is worth more.
House prices move much more rapidly than rents—both up and down—because of this investment value. The money that you put into the house can be put to other uses: you always have the option of renting, and either investing the cash elsewhere, or not borrowing the money in the first place. So house prices are affected by interest rates, as well as the rates of return on equities and bonds; the cost of borrowing and the return to lending or investing. When interest rates fall, house prices go up even when rents don't rise; when interest rates rise, house prices go down even though rents don't fall.
It is market rates that matter. Before the crisis many banks and building societies offered tracker mortgages that were linked to Bank Rate, and this is still true to a lesser extent. But they linked their actual market rates to Bank Rate because the BoE shifted its headline policy rate exactly when they'd want to shift their rates—in response to market conditions. When this rule broke down, during the crisis, many lenders simply dropped or altered their link. The link between the BoE base rate and house prices is weak, for this reason, but the link between market rates and house prices is very strong.
However, it's not just today's interest rate that matters: homebuyers do not repay their mortgage at today's rate forever; rates typically vary with the times. The unprecedented boom in London prices we are seeing today comes from the unprecedented fall in long-term interest rate expectations—the very same fall that many economists associate with "secular stagnation", the much-hyped idea that productivity growth is falling and can't be raised. And an infinitesimal fraction of private bank funding comes from the Bank of England, borrowed at its base rate: there is little reason to expect that future Bank Rate rises will change this. Instead, it is real market factors, like increasing supplies of savings coming onto the world market from developing countries, and/or poor alternative investment options, driving this expectation of low rates well into the future.
Interest rates—and expected future interest rates—combine with the supply situation—and the expected future supply situation—like two blades of a scissor. If planning rules are loose then a shock to the path of interest rates has little effect on prices: builders just build more to accommodate demand. House prices didn't fluctuate with interest rate trends before British planning rules became much stricter with the 1947 Town and Country Planning Act, and they don't fluctuate much in Houston or Japan, countries with much more liberal regulation on construction. But since we made building hard to do, we've seen large swings in UK house prices with market conditions and rates.
Expectations usually bring tomorrow to today. Unless we keep being shocked by the planning situation—it either gets stricter, or it stays the same but we expected it to get more liberal—then tight supply constraints are probably not to blame for rising house prices. Unless rising demand from foreign and British migrants continually shocks us that probably isn't to blame either. Those both contribute to the level of prices, and, sure, this involves an adjustment period, but most tenancies are not that long, and there are a lot of housing transactions, so the stuff we've known for decades is most likely capitalised in.
Quantitative easing isn't a likely candidate either, except because house prices and rents are higher when we avoid massive economic depressions. Money from quantitative easing doesn't "go into" things in a special way any different from other money. Investors don't suddenly change the fraction of their portfolio they want to hold in property when the Bank of England increases the money supply (slightly). In fact, we have strong evidence of "portfolio balancing"—when gilts become more valuable because of QE, investors sell some, and buy more of other assets to keep their total portfolio held in roughly the same way.
No: QE, immigrants, and even supply constraints probably aren't to blame for recent rises. Rises since, say, 2012, are most likely down to continual surprises in financial markets—expected interest rates trending further and further down than we've expected.
*That's not to say there's a free market: we have stamp duty land tax, council tax, business rates, housing benefit, affordable housing requirements, deep thickets of regulation on what you can, can't, and must do with your property, especially if you rent it out, as well as restrictions on how you can transfer and own properties, not to mention planning restrictions, building codes, and design rules. But, within these constraints, supply and demand interact to provide houses, flats, and rooms to consumers.