A very good-looking garniture!
Since these 'slate' cases are really a kind of black limestone, the Meadows & Passmore 'slate' blacking compound is designed for this, and works very well on a cleaned surface. They would have been polished originally in several stages, and the abrasives used would have depended on the exact type and origin of the stone.
One way of cleaning the case is to use pure turpentine, (not the substitute or white spirit), on a pad to remove the dirt and old polish. If it's really filthy, use a very fine grade of wire wool with the turps first, and if you're not sure, try a spot on the back of the case to see how it comes up.
The book 'The French Marble Clock' by Nicholas M. Thorpe is the best source I know for this detailed information on these fine clocks.