You’ve seen the house whilst it was having work done, it will look awful at that stage. Don’t panic!
Loads of people take ages to sort their house out, and most people who buy have had to put up with things looking grim when they first move in (For a few months at least!). Your family will have a secure long-term home at an affordable rent, and that’s way more important than every bit being “done” before you move in.
Priorities need to be the garden (for your youngest) the youngest ones bedroom (as the older one can understand he’ll need to wait till you can afford it), and the living room (as it’s a room you all use).
The fence - council likely to fix. If not a diy job will do. The rest of the garden, just do it yourselves. A couple hours a day at a time will get it sorted. If you can, borrow a strimmer (failing that, hire one for a day) and cut the whole lot (other than shrubs and trees, obviously!) down to start with. That’ll get it useable, and you can get the rest done over the winter. Most people can’t afford to pay someone for gardening work.
Walls - will pollyfilla, sandpaper and white paint be enough to make it OK for a while?
Flooring - personally I’d do your youngest sons room and the living room first. Bedrooms in cheap carpet. Get underlay at same time, and even rubbish carpet will last several years. Both rooms are likely to be relatively cheap (nowhere near £1500).
If you can afford it, find a big second hand rug for older ones room to tide him over, also a hallway runner. But only if something very affordable (free?) as is temporary.
If the kitchen is bare concrete, then you can lay vinyl yourself (can be cut with scissors). I would guess a £10/metre budget (including glue etc) is pretty generous for that. It doesn’t need to go under cupboards, just take off the kick-boards, run it to the legs of the cupboard then put the kick-boards back again.
The cost your quoting for blinds seems v high.
Hope you enjoy you’re new home!