To ask how single SAHMs survive?
Well, either they have a partner who financially supports them, or they live off benefits, or they work.
In your case, you don't have a partner.
You also don't appear to be able to live off benefits, although a good proportion does. (Don't know if you're not entitled or not happy with the income level this would provide - it's moot anyway, the outcome is the same).
Thus you're going to have to support yourselves.
You have a professional degree - yet seem to be spectacularly unrealistic / uninspired about what kind of jobs you are perfectly able to do if part time, term time working isn't going to pay your bills (just like the rest of us).
You are in a better position than a good many parents who find themselves supporting a special needs child. Use some of your years of child-focused training and set up something related - people have already mentioned childminding and tutoring. But what about other things that are somewhat related?
There are many jobs which can offer some degree of flexibility if you are a good employee (even if you can't arrange that part time contract/flexi working immediately at starting). How good is your English? You have a degree - how's your essay writing skills? Is it strong enough to look into copywriting or other written work? Do you play an instrument - is there any other skill you have that you can use to make ends meet whilst caring for your DC? This is what you need to focus on. Not "non employer around here will pay me £30k a year to work four hours a day during term time only" - which is essentially what you're hoping for. Get over the fact that it doesn't exist (unfortunately!).
Also, perhaps seeing if there is some sort of local support network for parents in your area?