I once spoke to an estate agent about this, because the question puzzled me too. He said they do it in small steps, often starting with a shared ownership arrangement and then selling on, taking advantage of property inflation and low interest rates, and their own increasing income. Anyway he felt the real key is at the start, because as you say the deposit is a problem, and a shared ownership of a small flat makes it realistic.
London is a huge problem I know. Young couples both teachers who want to stay here, and who are earning mu money, find it hard to buy a house big enough for a family with two or three kids - despite decent income and a stable job.
By the way, the rentals market in London is crazy at the moment. I’ve never known it like this. It’s a very bad time to be looking to move to a new rental here.