Hi Jim,
I would be very careful using certain cleaning products on this type of clock case. Krud Cutter can be very harsh too, I would avoid using that as well. I would start with using a bowl of warm water with very little mild dish soap and a clean wash cloth. Then dry off any damp areas. Don’t apply any mineral oils or waxes until it is completely dry. I can almost be sure from just the look of the pictures that this is just superficial dirt and should clean easily. I clean most all clock cases with this method (depending on the condition) and had no problems or damage to the finishes. For the groves and tight corners, I will use a soft tooth brush and Qtips. 
I wish you well
Jim