But sheiks and oligarchs are buying all the real estate, so the cost of living is pretty similar.

In central London perhaps. But it's by no means important to live actually in London and work in London. Commuter culture is (or at least was) big. Many people commute from the surrounding 100+ miles by train each morning and pocket those London salaries while living in low cost of living areas.

And some part of them slowly dies each morning when they have to wake up extra early just to waste time on train delays.

Commutes over 30 minutes extremely sucks, unless they are taken from your work day time.

if you live in the fucking shard, maybe. london is expensive but let's not exaggerate: a £150k earner is not worried about the cost of living.

Bigger demand for higher-level housing creates price pressure on all levels.