Putting a chain back on a cuckoo clock's gear is possible but not easy to do. If, however, you do not have access to a clock repair shop that specializes in cuckoo clocks with some time and patience you can make this repair yourself.
Pull all the chains all the way up, remove the weights and carefully tie the chains securely together at the bottom of the clock so that they do not move at all. This is necessary since you'll be turning the clock upside-down.
Turn the clock around and remove the back. Look inside the housing; you will see the gears and your fallen chain caught on one gear' s axle. Your goal is to thread this fallen chain back onto its gear.
Stick your hook inside the clock and find out which direction the gear turns by gently trying to rotate it in either direction.
Turn the clock upside down and use the hook to work the fallen chain up and over its gear. It might help if you can turn the gear with your finger as you gently pull on one side of the chain to help it to catch. Once you have balanced the chain up and over the gear, pull both ends of the chain taut and carefully turn the clock back upright.
Hang the clock back on the wall, untie all of the chains and re-hang the weights. Pull the weight on the chain you just fixed all the way up to match the other weights. Set the pendulum in motion; your clock should now work properly.