You CAN work, though. It's not impossible at all. You need to get past that way of thinking and start looking at other pathways.
I was in a similar situation to you, as said previously.
You NEED to tell the school that enough is enough. Unless it is an absolute emergency (i.e. your child is in danger to himself or others and the school has exhausted EVERY option to remedy things themselves) then you will not be collecting your child until hometime. Start making yourself busy during the day. I bet that they're calling you so often because they know you are so readily available.
Why on earth have you not applied for DLA already if your child's needs are so significantly affecting him at school? If you get the school to write up a short report then that will be considered a very valuable piece of evidence by DWP. You do not need to await diagnosis. The waiting time for first applications just now are about 3 weeks. My child was awarded HRC and LRM before diagnosed.
It is based on NEED not diagnosis. E.g. someone with Down's Syndrome may be functioning just fine in life and requires little to no additional support despite a diagnosis. Someone else without a diagnosis may require a shit load of support and therefore more likely to be awarded DLA.
Can you study just now while you're at home, improving your options for work later on when the youngest is at school? Have you looked at the Open University website or checked if your local uni does distance learning modules? These may require you to attend in-person lectures maybe just once or twice a year.
You'll get lots of your nursery bill paid by WTC if you take on a job over 16 hours. It's difficult nowadays to find something with a set shift pattern i.e. Mon to Fri 9am to 3pm. Most things require you to be 'flexible'. Enhancing your skill set through volunteering and studying should make you more attractive to potential employers for the regular hours jobs. I really struggled to find something that didn't need me to be flexible or have a driver's licence (didn't learn to drive until last year). That's why I started to do some self-employed work. Ironing and cleaning. I would take my little one with me. Zero start up costs. And I just spent a day on the business gateway website to get my head round keeping books for the business etc. If you get FIRM with the school in regards to your eldest, would self-employed work (taking your youngest with you if nursery placement is problematic) be achievable for you? Or taking in ironing? I know it's a cliche, but I was able to come of IS after four months of starting up cleaning and ironing.
Acquiring paid work when you are a lone parent and have a disabled child IS POSSIBLE. It's just much harder than if you had family support and a child who can attend ordinary childcare options. You just need to think outside the box as to how to jump around all of your obstacles to find a job that fits in to your life.
I know it's hard to access support groups when you are a lone parent (why the hell are they always at night time and for adults only). Can you look at joining a forum or the SEN boards on here for some virtual support? It really will make so much difference having a safe place to rant and feel listened to.