My neighbor came over yesterday with a classic problem...a board too short.
He was repairing it for his sister and cut the coopered top 1/2 inch too narrow. He cut it to fit the opening, but forgot to add the part that goes into the groove.
Anybody have a way to stretch it ?
Work up from the same wood a new board identical to the too short slats. Cut off 1/4" thick slices; 2x the number of slats. This may require hand cutting on a miter box using a sharp saw. Glue on the new end caps and pin nail for support.
A gluing hack: because you're gluing end grain to end grain, glue up in 2 stages. First stage: dilute wood glue with water and apply to end grain surfaces; let it dry. Second stage: apply wood glue to surfaces, set, and pin nail. The resulting joint will be almost as strong as face grain to face grain because the diluted first coat soaks into the end grain much deeper.
The seams should be almost completely hidden in the grooves. It will probably take a pro to spot the fix. At minimum it will be less noticeable than a glue line right down the middle of the roll top.
My suggested approach varies slightly, depending on the answers to a few questions. A photo would help.
Does the tambor have interlocking (sorta ball and socket) slats or are the slats joined the traditional way with canvas glued to the back. I assume, if it has a canvas backing, you don't want to remove it and totally rebuild the tambor if you can avoid it. Are/were the ends of the tambors that ride in the track thinner than the tambor (tenons) or were the ends the same thickness? If the ends were tenons, was the tambor "shortened" from both ends or just one?
If the resulting too-short tambor/tambor slats fit the inside dimensions of desk sides so there is minimal gap and neither end is a tenon (slat with reduced thickness) that rides in the track, make a simple drilling jig and use wood dowels to attach extensions to each end of the tambor slats. If the tambors are narrow, you'll need small dowels. Depending on the fit of the shortened tambors you should not be able to see the joint between the tambor and the extension.
Instead of adding extensions with dowels, just use large dowels whose OD is the same as the thickness of the slats. Reduce the OD of a 3/4" or so section at one end of each dowel so it can be inserted into a small dowel hole in the end of the slat.
Another, even simpler method that requires no turning would be to glue appropriately sized wood dowels or even plastic rods into the ends of the slats. Then, over the dowels/rods slide plastic bushings whose OD is the same as the width of the track. This method should result it a tambor that slides easier than the original.
A slight varitaion of the two previous suggestions- like them, add wood or plastic dowels to the ends of the slats, but reduce the width of the track by adding a strip or two of wood or plastic to the inside, so the track is just slightly wider than the diameter of the dowels. To maintain the same placement of the slats in the track (if the desk has a mortise lock), you could also insert a single strip of hardwood or HDPE/UHMW-PE that is the exact width of the track as a track liner. Route or dado a grove that is a fraction wider than the diameter of the dowels/pins. Since it will be visible when the tambor is up, you will want to stain it to match the desk color or paint it black, etc.
Any method requiring drilling dowel holes in the ends of the slats will require a drilling jig to insure all holes are drilled straight and in the same position.