Repairing Cuckoo Clock Bellows
Below is a good overview video from James Stewart demonstrating how to repair cuckoo clock bellows.
For antique or higher value clocks, the original clock would have had actual leather bellows, and you’re going to want to retain the leather so as not to affect value. Leather bellow material can be obtained from Columbia Organ Leathers. For newer/vintage clocks people tend to use Tyvek paper, which can be obtained at any horological supply house.

For cuckoo bellows I purchased the CPL Thin (.010″-.012″) sheep skin material from Columbia at $16 for a 6″ x 12″ sheet + s/h, which is enough to cover roughly 3 standard cuckoo bellows and 2 smaller quail bellows.
Note that the Columbia Organ Leathers website is currently under construction, so you will need to call or email them to order. Aside from this hassle, the process went smoothly and they were a pleasure to work with.