I’m a Londoner and I completely sympathise with you op.
Through busting our arses saving, DH taking a second job, and a very generous donation from a relative, we managed to gather together just over £100,000 for a deposit. We still had to take out the absolute maximum amount of mortgage available to us, and then the cheeky fuckers told us we’d have to pay off our student loans in full in order to give us that amount! We managed in the end to do this.
In the area we had been looking to buy in (and eventually bought in), the prices for the type of flat that we bought almost doubled in the space of 3 years whilst we were saving. It was a nightmare situation, thinking that we were getting somewhere with saving, then for the astronomical rises to keep detracting from our savings.
I do know of some areas in and around London with good transport links and schools etc, with properties in your price range (depending on what kind of property you need). We’re not living in one of those areas, as they were too far away at the time from our workplaces, and DH was being a bit too snobby too, but is now realising the error of his ways!
Private message me and I’ll give you more details of areas if you wish.
Good luck, and well done on saving such a large amount.