Advice desperately needed

[custard]

Would I be correct in saying that the lid and base would/could be the first thing to do in that the veneer would alter the thickness requiring a wider slot for it to sit in?

Absolutely correct. Not only will the veneer vary in thickness, plywood (even top quality Birch ply) is notoriously variable in thickness, MDF tends to be more consistent. So the smart worker veneers first and only then cuts the grooves to suit.