I just want to preface this by saying I'm not asking for anything other than practical advice. It's not a begging thread 
I'm a single parent of two small children and I live in London. Due to unforseen circumstances (bastard ex) I've found myself in an undesirable situation.
Me and my two DC (ages 1.2 and 3 years) one of which has special needs are living in a box flat which consists of a living room with a kitchen built in to the room and one small bedroom. No outdoor space. Plenty of faults with the flat. It was all I could afford at the time of needing to move urgently but has become very unsuitable as the DC get older.
I work part time and am unable to take on any more hours until my youngest is eligible for the free hours when they reach 2.
My rent is £1,000pcm for something as small as this which I can afford as I have my wages topped up with UC, but we need to move because the flat is becoming hazardous for my child with SN.
I can no longer contain him to a travel cot and living in what is basically a kitchen with a sitting area is dangerous. I've child proofed the place as much as I can but it's just not suitable and is taking it's toll on my mental health being confined in such a small space.
I'm not eligible for council accom as I haven't lived here long enough and there isn't anything anyway, waiting lists are years long.
I could save for some rent in advance for a new place if I was very frugal for a few months but all the estate agents want so much money, rent in advance on top of a deposit and then admin fees.
It cost me 2.5k to even get over the threshold of this shit hole of a flat and this was supposedly a bargain. Private landlord who has no other properties.
I can't leave London as i rely heavily on my support network.
Are there any charities that could help IE affordable housing or organisations that support people in my situation in finding appropriate accommodation.. or am I on my own? I can't see a way out and the council aren't interested as we're housed 