I don't want to completely surrender to the magical arts to solve everything, but I would assume preservation magic would be a key component to the craft as would techniques for mending; An animated body would still be accident prone almost to comical levels. Without the ability to mend these common wounds and defects a servant's usefulness would be fairly short lived. As mentioned previously I would assume the system is already animated almost completely by magic, having removed both circulatory and digestive systems. This would suggest that the body mechanics were being powered by magic anyway, so why wouldn't that same essence keepskeep things preserved, at least, and possibly regenerating.
A step back from that would be to go with a variation of the earlier solution and create an alchemical cream that the zombie is instructed to apply daily before and after work. The cream would basically be a bridge between the existing magical energies and the dead cells of the servant. It would be the zombie equivalent of a transplant patient needing to take anti-rejection meds for the rest of their life.