@LaFemmeNicola that is an awesome achievement and good on you for exploiting that field when it was most lucrative. With no family money and in the absence of a posh or monied background, that's really unusual.
OP I'd say focus on what you actually want out of life. Is it actually your current career, social circle, and location? If so you're going to have to just suck it up I think, and either live in a small unappealing property for a bit or continue renting indefinitely. It sucks but it's the way it is. Look at Tokyo and how families live there. It's not unusual to live in tiny spaces.
I agree with looking at places like: Mitcham, Stoneleigh, Morden, Croydon, Carshalton, Chessington, Epsom, Penge, Feltham, Hounslow, etc. Not lovely but perfectly liveable and with good schools. That's just west/south west, there is a whole ring around London of outer suburbs.
You can get an e bike or moped to save on transport costs longer-term and if you are commuting in daily.
Make sure you are claiming absolutely everything you are entitled to as well. 30 hours free childcare as your child grows, tax free childcare, cycle to work scheme etc. Sell what you don't need. Ditch car if you have one.
Too good to go app for food, lidl £1.50 veg boxes, etc. It truly sucks but your (and my) income isn't very high for London given costs. In cash terms it is great but it does not go far.
Also could you take up a second job or sideline? Translation, audio typing, matched betting, some sort of consulting allied to your job? Again, this is the reality for my peers (and me) right now.