There was always going to come a point where born and bred Londonders could no longer afford to stay living in their hometown, and if the fact people with higher housing priority than you are sat waiting for years before being offerred property isn't enough of a deterrent, then you must surely realise your only real option is to look for private rented housing, using a Guarantor or the council bond scheme.
Instead of just asking at letting agents or checking Rightmove, you can still search for flats the old fashioned way by looking in shop windows, supermarket community sale boards, asking around who are the local landlords (ask your local corner shop, they know everyone!).
I also once found a flat just by posting a note on the staff room board, although I appreciate that option isn't open you being currently unemployed, you might be able to ask the receptionist at some of the very big local firms if they have such a board you can check.
Just because you may well see a similar young woman with a baby in a council flat in Ealing, you have to shelve the fantasy that will be you anytime soon. She may well have sat it out in grotty B&Bs for several years before being offered that. She may have come straight from prison where they're instantly classed as priority homeless. She may have invisible disabilities that prioritised her. She may be subletting off the actual registered council tenant who lives somewhere else.
It's a shame you can't get to know these people at playgroups when baby is old enough and see if they're willing to advise you on how they obtained their property.
Ultimately, although it seems overcrowded and stressful and not at all ideal, living at your Mum's will be far preferable to miserable years in a council bedsit B&B. Go and check such places out if you don't believe me.
You're in one of the most expensive capital cities in the world. Yes parts of it are so crud-u-like you can't imagine why these a housing shortage, but the reality is people are happy to live in London's crud-u-like regions, rather than move out of the city altogether. Where you are now, at your Mum's, with your current unemployment and family status, is the best you're going to get.
You are nowhere near being 'priority homeless'. You're not even homeless. And overcrowding is no longer a priority factor in real terms.
Stay there, or consider starting your life afresh in another part of the country.
Move to West Yorkshire or East Lancashire. You'll get offered a 2 or 3 bed house instantly as they have 'instant lets', or you can wait a couple of weeks for a nicer area home.
There are some lovely towns up that way. Todmorden for instance. Hebden Bridge.