In my experience, most clock cases do not need to be refinished. Through the years they accumulate dirt and grime. Through special cleaning and polishing, most cases can be brought back to life, without removing the original finish. When case parts are missing, I have several sources of suppliers who specialize in junk antique clocks that may have the original case parts needed. Missing wood case parts can be reproduced.
Clocks dials can be restored or replaced. A paper dial is the easiest because replacements are still being printed of most antique clock manufactures. A painted dial can get very difficult to have restored or repainted. Modern clock dials are available for most German movement.
Reversed painted clock tablets (glass) can also be restored or re-painted.