It has been a while since I repaired a tambour sp? As I recall I removed all of the canvas from the slats. I then removed old glue somewhat. If it was hide glue, vinager should help out on that task. I cut a new piece of heavy canvas to size and glued the slats back down. The trick is to get all of the slats glued down and 1. not get any glue on the last 1/2" of each end. 2. getting all of the slats in line so it is not overly tight in its grooves when done. 3. get all of the slats close together so you dont see a lot of canvas when you are done. I think the method I used was to put glue on both the canvas and the slats (a little at a time so the glue doesnt dry) unless you are using contact cement then it doesn't matter. Have the canvas stretched out and tacked down to a piece of plywood. Canvas should be larger than you need in all directions. Draw lines where you want the glue to be. I used contact cement. Hide glue would probably have been a more historically correct choice but this was a paying gig and I knew that I could (not) pull it off so to speak with contact cement. Lightly clamp the slats at ends so all of the ends line up. I don't think you'll have to clamp across the long side of the slats. The stretched canvas should recoil a bit to close some of the gaps.

good luck.