I just finished up a project on my TR3, where I covered the glovebox door with a walnut burl veneer. I know, I know, it's not original, but since I'm going with a wood steering wheel, I figured it would be a nice touch.
I found a supplier, veenersupplies.com (I think) who seemed to have a real big selection in terms of available sizes and wood types. Plus, they have photos of what your buying too.
I just laid out the door on the veneer, traced and cut it out with a grind off wheel with my Dremel tool. Cut it so that it is about 1/16 too big around the edges. Used basic contact cement to adhere it to the door face. My door was small enough that I could clamp it up in my vice with some C-clamps too keep it tight. Sandwich using wastewood so you don't leave clamp marks. I imagine you could do the same with a bunch of c-clamps and plywood. I did notice that using the contact cement that the veneer bubbled in a few places, which wasn't a big deal, you just need to take roller and reset it.
Next I sanded the edges down flush with a fine sandpaper. Don't use a dremel here because the veneer is only 1/42" thick and that will probably cause more damage. You'll see that sandpaper will take it down fast enough.
Next, I then took glossy polycrylic, thinned it by about 50% with water and began applying it to the veneer. The reason I thinned it was so that the grains would become fully embedded and clogged with the poly, and so the poly would not just merely just sit on top of the wood. You will see that the wood will suck up the thinned poly. I used a foam brush for this stage, and very lightly sanded between coats. After you see the grains completely clogged with poly, then start spraying with a spray can of glossy poly. I put about 15 thinned coats on and 5 spray coats.
After it has completely dried, I buffed and waxed with automotive wax, then cut out the key lock hole. Make cutting out the holes the last thing you do, I think you will find it easier than having to work around the holes. To cut out the hole, I used a tiny little Dremel router bit that is available. It worked great, and turned out looking fine.