The next big question was what wood to make the laminate from. One option would have been to make all but one layer from something fairly cheap and use something fancy (e.g. a pretty burr) for the visible outer veneer. After much searching and calculating and debating (with myself) I opted to get a long strip of Indian rosewood and use that for every layer. That way the end result should look fairly close to the original, providing you don't peer too closely at the edges. I could in theory veneer the edges too, though the shape might make that rather tricky and it would still be possible to see the laminations on the inner edges of the fretwork piercings.
Here's what the frighteningly expensive rosewood veneer looked like when it arrived, rolled up into a tight cylinder.

After unrolling it, with the concertina in shot to show the scale.

The first stage of making the laminate is to cut the veneer into rectangles (six per board), half of them with the grain running in the opposite direction, and then flatten them so that the pieces will glue together nicely and I'll hopefully end up with a flat board in the end. Incidentally I tried cutting the veneer with sharp scissors but it was causing it to split along the grain so I reverted to using a very sharp knife and steel ruler. I'm using a veneer softening liquid which is a mixture of water, alcohol (meths), glycerine and hide glue. I warmed it very slightly to avoid the glue thickening. It has an 'exciting' smell! You soak the veneer in it until it goes floppy, then dry the pieces for a day or two between layers of scrap paper (replaced every few hours) and flat weighted boards (I'm using offcuts of kitchen worktop). I tried the technique and potion out first on a small piece of relatively cheap sycamore veneer that I'll be using to put decorative inlays into the corners.
Here are the pieces of one board, freshly soaked and about to be pressed:

Time spent so far on cutting and flattening the veneers: about 3 hours.