That is what as known as a “Postmans Wall Clock” , would have been made in the Black Forrest in Germany were they make the Cuckoo clocks. Yours is an early one I think as it has the painted wood dial. They can be held together with screws, wood pegs, tacks, all sorts of things, it depends on who worked on it in the past. In the wood plates you will find brass bushes, sometimes they are split bushes, they are cut along their length so when they are inserted in the wood plates they open out slightly to help hold them in place. Treat the whole thing like a normal clock but dont put the wood through the cleaning machine.
Paul.