Everywhere has an address and postcode. So whether you are camping at a campsite or on private land, that is your address you can register from. I'm assuming you are not wild camping for some reason? If so, register at a friend/relative's address but that is hardly a satisfactory ongoing situation.
We live in Turkey but I'm coming back to UK to
Work and we also have dogs , it will be hard to find
A house to rent so thought of caravan as an option , but on a proper site ?
Could be difficult to find a site allowing you to stay more than 28 days and not that many stay open all year. If you know somebody with a farm you might get away with it but for permanently sited caravan planning permission will be required. It is possible to do it but you need to be a bit savvy and inventive.
its not hard to find a house to rent even with dogs, they just charge you more on the deposit that's all, also finding a site that will allow you to live in it full time is going to be harder than finding a house
There are sites that allow caravans to be sited permanently. Admittedly they shouldn't, but they are out there. It is just a case of finding them, or asking sites in the area you are going to be staying in if they would let you. If the worse came to the worse, you find a couple of sites you like and move between them every 28 days. Hopefully not to far from your child's school.
As far as school registration goes, you would just have to use a site address. You could always ring schools in the area you are going to and see what their policy would be.