In desperate need

The picture is me and my boy. That's ... Us. 

Long story, and I'm actually embarrassed about asking for help, but I'm out of options and this is my last resort. 

I put it under this heading because I wasn't sure what else to put it under. 

So ... 

My mum was diagnosed with stage 4 leukemia in February of this year. Of course we all know, stage 4 means it's terminal. They're on a rescue mission at the minute. It's in her blood, lymph nodes and covering over 80% of her skin. 
She's just finished a gruelling 6 months chemotherapy course. 
She's currently having skin light therapy. 
Next week her radiotherapy starts, and then to finish it all, she's having a bone marrow and stem cell transplant. 
I was made redundant from my employment while I was on maternity leave 14 months ago, and when my mam was diagnosed I decided to put off going back to work because I was fortunate enough to be able to spend all the time I possibly could with her, through her good days and her bad. I don't regret that. I'd do it all again and be broke if it meant I could be with my mam. 
Now I'm in a position I can't get out of and it's killing me. 
My mum has her transplant coming up. With her transplant, it's going to leave her with a critically compromised immune system. It's already compromised anyway after all her treatment, but it's basically going to reset it. She's been told if she contracts infections within this time (6-12 months recovery period) it could prove deadly as her body won't have a "mature" immune system after the transplant to fight them off the way ours normally would. So, even as her consultant has advised her, she needs to have extremely limited contact with my boy, so we need to move out asap. 
I applied with the council nearly a year ago and submitted multiple change of circumstances to inform them about my mum's condition and that this was coming, and that me and my son would have to leave. They haven't helped. And I know, trust me I do know there are so many people who need houses and so few houses out there to put them in, and I'm not saying I deserve them more than Joanne up the street with her own story to tell, but I'm out of options. 

I had to go to a homelessness meeting last week and spoke to a housing officer there. We went through different options. I don't have any other friends or family I can stay with. I'm not leaving here because I want to .... But I need to create an environment for my mum where she can rest, recover and try beat this thing. She can't do that with my baby in the house. And that absolutely breaks her heart on its own. I can afford private rent but can't afford the bond/deposit and first month's rent that they require. 
When I spoke to my housing officer she told me about the "homelessness prevention fund" and told me that if I found a private rented place, they'd use the fund to help me with the bond and 1st month's rent. 
I spoke to so many letting agencies and explained my situation, and every landlord who was "DSS excluding" has accepted my circumstances and agreed to let as long as I can pay the bond and meet the remaining letting requirements. 
I called my housing officer to use the homelessness prevention fund and she said "oh, sorry, I shouldn't have really mentioned that because the fund is empty, there's nothing there to help you with" ... So there goes that idea. The conversation didn't end there because I, of course said, why did you mention it and put so much emphasis on it if it's not even an option for me, to which she said that she didn't think I'd be able to find anything in that sector. There's no help there financially. I can't apply for a discretionary housing payment to help me either, because I'm still at my parents and not currently in receipt of housing benefit. 
The only option I have now is temporary accommodation. They keep telling me that it's on a "what's available on the day" basis and keep saying that it's likely it won't be local, and could be anything from a hostel to a b&b ... Again, whatever is available. I could do this for up to 6 months. 
I know beggars can't be choosers and I'm so sorry if this causes any offence, because I can't stand the idea of them sending me miles away where I won't be able to easily get trains or buses to my mum. Or into a hostel where my baby is around strangers. But unless I can magically find £1500 for my deposit and first month's rent within the next 2 weeks, this is what I'm going to have to do. 
Please don't think I'm being a snob or anything, it's just not what I hoped for for us both. Hopping from hostel to hostel struggling for somewhere to sleep. He gets so unsettled in strange environments anyway and I'm absolutely dreading it. I hardly sleep a wink for worrying about stuff already, let alone laying in a bed having no clue who's on the other side of the door where me and my baby are sleeping. 
I know I should be greatful they're going to help me with at least a bed to sleep in. I know. I'm just dreading it. 
With my mum fighting this, finances are tight anyway, so there isn't even anyone I could go to even if I was to ask for help for the private bonds and stuff. 

So this is my last ditched attempt to secure a private tenancy before I have to go down to the homeless office. I don't expect anything really, I don't even anticipate anyone will make it to the bottom of this, and bravo if you did. I do waffle on a lot. But if you did, thankyou for reading
Everyone has their rough patches in life, I guess this is my rough patch. It can only get better, huh!