Hi K
Rolltop eh? Imade a rolltop breadbin many years ago. I used paranha pine - big mistake, it stank to high heaven and tainted the bread. Binned it soon afterwards!
However, what I learned from that was:
1. The grooves at each side have to be IDENTICAL mirror images. Make a template jig.
2. Don't make the radii any tighter than absolutely necessary.
Traditionally the tambours themselves aew glued to a canvas backing, but I can't remember the details - presumably animal glue is the trad adhesive. There are better historians than me on here, so maybe one of them can help you further.
Best of luck with it, sounds like an interesting project.
Cheers
Steve