What is a good cost effective material to use for an outside build? Could MDF/plywood be treated?
Plywood and regular timber are usual. MDF soaks up water like a sponge, as does chipboard, so neither are recommended for outside use. Any wood you use outside will need plenty of protection if it's to survive more than a couple of years.
This can vary from almost zero to a couple of grand depending on materials. People have built boxes using timber and plywood sheets from skips that look pretty respectable. It depends how much you're prepared to scavenge. If you buy it new, 5 sheets of plywood will cost around £125 (if you're in the UK), and you'll probably want quite a lot of 1x6 for the corners, and 1x3 or 1x4 for decorating the walls and doors, which will likely cost more than the plywood. Top signs, door signs, handles, hinges, and lamp will all add to the cost. You may also want to use acrylic 'glass' rather than real glass if the kids will be using it.
If your TARDIS is going to stay outside, you need to think of it as a shed, and prepare the ground for it, as a full sized TARDIS will normally weigh a lot. You'll also want to have a treated timber base where it touches the ground, and stand that on concrete or patio slabs.
A smaller TARDIS is cheaper and more portable, so it could be moved into the garage when the weather is bad (wheels are good!), but the kids will also grow out of it faster than you might imagine...