It sounds like you have a very bright little boy, possibly with SEN, who needs to be out and doing, to be learning, yet he is home most of the day, bored and without play mates.
If you are home all day, how have you replaced his lessons? Do you teach him reading and basic maths? Are you helping him with starting to write. Does he have a new reading book from the library each week?
Where do you take him during the day? Woods, canals, museums, What about sport? Junior ParkRun is free. Can you start to teach him basic food prep - how to make a salad. Can you buy a tomato plant or two from b&q and show him how to grow stuff. You only need a growbag and some plant fertiliser.
Until his schooling is sorted, you need to fill his mind and keep him busy. Tire him out. What about regular dog walking for the dogs trust or a neighbour?
Does your partner take a turn? It would be a break from job applications for him, a bit of peace for you and a good role model for your son. Can he take him fishing or playing kick about. Maybe you could write a basic weekly activities chart to create a bit of routine and to help your ds know what to expect. The more you can do, the easier it will be for him to return to regular schooling.
It sounds like you are all going through a difficult patch. You are doing a good job coping at all. What is the local council doing about schooling him? They have a legal duty to provide for him, and should be helping you.