Can I use a water jet machine, CNC or table router to cut mica lampshade sheets? Any advice would be appeciated.

They all sound like overkill to me. I use either a utility knife or heavy shears, and then you can use a file to smooth the edges. It doesn't take much to cut through even the thickest sheet of mica; a handheld jeweler's saw would be even more precise if you need to cut an intricate pattern.