Thanks everyone - incidentally, that link I meant to put in last time I missed out, so I'll put it in here so I'll know where to look for it and in case it's handy for anyone else.

The back is rounded rather than bowl-shaped - it's not a Italian style. Lie it on its tummy, and it's a pear-shaped shallow dome about an inch or so high, resting on walls which are about three inches high. There's not any kind of joining strip (purfling) where the tringles lie on tye walls.

The back is made up of triangles - well, they aren't triangles, because the outside edges are curved, and each one on either side is a slightly different shape, so that they fit on to the side wall; there aren't any braces on the back, but where they come to join at the centre they fix onto a bit of wood inside about two inches square - the whole dowl acts as a brace for itelf it seemes

The triangles are alternately plain light brown, or they are made up of 4 smaller pieces in contrasting dark brown and light brown, each meeting at a point in its middle, a sort of saltire shape. Fortunately these pieces are all intact.

The 5 triangles at the machine head end are properly in places, but the others are either loose, or need to be re-seated. At all the edges the triangles meet each other (including the "saltires" I mentioned, there's are black lines which I imagine is some kind of inset veneer or somethoinhg of that kind.

Hard to describe this kind of thing. This is where the virtual world is frustrating.

Maybe what I need to do is find an evening class in this kind of thing, with the double aim of getting the skills specifically for fixing this one, and of having the skills, so I can get into making and mending other instruments. But there don't seem to be any classes like that in range. Or chase around for a mentor.

So I suppose the things I really need to know are
1)how to set about removing the pieces from the mandolin, and cleaning off the old glue, without causing damage, and
how to build a frame on which I can reassemble them and
3) how to carry out the reassembly, including adding in a few little patches especially where the whole structure comes together at the centre, and replacing some of those black veneer lines or whatever.

The whole thing looks like something which would be fairly straightforward to do well; and very easy to totally screw up.