Note: I haven't seen one of these systems before, but it appears from the technical documentation that the common point is the battery holder; the front and rear derailleurs are not otherwise interconnected. This should narrow the list of things to check.

Inspect the wiring and connectors, especially at the battery unit and the shift switches. Look for corrosion or damaged insulation. Gently pull on each connector to verify it is seated and latched. Where the wiring is subject to repeated movement (especially at the handlebars), flex it and observe whether it bends smoothly. A broken wire inside the insulation will let the insulation turn more sharply at that point. Breaks are most common near where wires are clamped down, so look closely at those points.

If the wiring and connectors look good, inspect the battery contacts for dirt or corrosion. Remove dirt with a soft cloth wrapped around a dowel or pencil eraser. Use a clean pencil eraser to remove tarnish or light corrosion from the battery contacts. If the corrosion is too thick to remove readily with a pencil eraser, cut a strip of Scotch-Brite and fold it across the end of the pencil eraser (make the strip long enough so you can hold it to the sides of the pencil), then use that to clean the contacts.

If you haven't found anything in the preceding work, put the bike on a stand and get a helper to turn the crank and operate the shift buttons while you gently push on the wires going into the battery unit connector. If one of them is broken, you may get intermittent operation. Don't overdo it; you may cause a break!

If you have a multimeter, you can disconnect the wiring and check it for continuity; this is easier and more reliable than push/pull tests, and will not put so much extra wear and tear on the wires. The next step after that would be to connect the battery unit, insert the battery and check for voltage at the shift switch connectors.

Shimano provides an electrical system checker tool specifically for this system, part # SM-EC79. This will not only test the various parts of your system, it will let you reprogram the shifting direction of your shift switches. I don't know how much it costs, but don't be surprised if it has a "sticker shock" price. If you are on good terms with a bicycle dealer who has one you may be able to borrow it.