Medieval carpentry is what you want. If you are doing this as an experiment in how much you can create with next to nothing, you probably already have all you need (although I would want a handsaw pdq, purely from an effort vs reward point of view). Become a bodger and make chair legs in the green, using a sapling and a piece of string. All sorts of things are possible.
If I had to make your shelter, I would use mud. Clay, to be precise. It depends on what you have available, so dig a hole below the topsoil and see what you have to work with. Cob would make you an pretty robust, permanent building, but would take time. You could knock up a wattle and daub wall in a morning, including cutting the poles. Lath and plaster is similar if you want to get fancy. As long as you can keep the rain off it will be permanent, and if you can't it will need some repairs from time to time. You have time.
Your first dwelling might be more of a stone age roundhouse, and then once you have a base you can expand at your leasure. Of course the government may well have things to say about permanent structures without planning permission, but I imagine you didn't vote for that, so what they don't know won't hurt them.
First on my list of things to make would be a shavehorse, so you have both hands free for your workpiece. A drawknife would be the obvious tool of choice for that.
Good luck. Well done for withdrawing from the system - if more people did it, there would be a lot less murder and mayhem around the world.
