Hi OP, sorry to hear about your situation and hope you and your kids are safe.
I'm in South West London so I know how bad the housing situation is. I've just been housed with my 5 month old after going through the homeless route with the council.
My best advice is going through the homeless route and to contact the council to say your partner is kicking you out and both you and your kids have to be out by the end of next week etc. You'll have to fill out a homeless application on the council's website (whichever borough you're in) and someone will get back to you to assess you. It's quite a long process and you will get placed in a hostel or if you're lucky like I was, you'll get housed in a self contained flat which is basically a studio flat. It took 9 months for me to be housed however there is no way you'll get a council property unless you're able to wait years in your hostel/studio until one is available.
Many councils have a scheme for homeless applicants where they're able to rent privately instead of being offered a council flat. The advantages to this is you don't have to pay a deposit as the council pays this for you, the council usually negotiates a lower amount of monthly rent with the choosen Landlord and that Landlord is happy to accept people who are on UC and other benefits.
The other option is to contact the council and let them know about the abuse. However this only really works if your abuse is known to the police/GP so there's professionals who are able to back up your situation. The council moves quickly when there's a situation with kids involved and a woman is facing abuse but again, they're most likely to place you in a hostel/studio to take you out of that environment and moving forward you'll either be advised by your housing officer to take up private renting or wait a considerable amount of years until you're housed in social housing. (I was told I'd have to wait approximately 7 years in temporary accommodation until I was able to be housed in a council flat
it's utterly ridiculous.) Either way it is an extremely long situation in which you won't end up with a council flat anytime soon anyway!
Another plus for taking a private property is your hosuing officer will move you from Band D to Band B on the bidding list in hope you'll get a council flat within a few years of living in your private property. Again this is unlikely and you could be waiting up to 5 or so years depending on your location.
Sorry for the long post but please feel free to personal message me on here for more information or any questions and I'd be more than happy to help