It sounds as though you have good advice but the Citizen's advice bureau do offer help and advice,,,, also the council might be obliged to offer you housing. It's all very well having the law on your side but if the landlord decides to be awkward then he can make life hell for you. I hope it works OK for you
I hope this works out for you--good luck on it! Let me kindly suggest, once you get settled, that you start stockpiling cash against something like this happening again. In the original post, you said you did not have enough saved up for the first month's deposit at another place. Aim at making that never happen again. Talk to someone how about best to arrange your finances so you can stick even a little bit aside each month by paying yourself first. It does not need to be an amazing amount--even a little will add up each month. Do it so that your four year old is never in any danger from something happening to you or your partner. Do it for your child. My two pence; let us know that you get settled soon!
Put your name down now!! Back in '81 in Cardiff I got made homeless through no fault of my own and the council gave me and my family a 3 bed council house the same day,,,,,,,,,, I guess things have changed a lot