This is true. Nerves work by separation of charges, not electrical conductivity. Charge separation is still electricity, though. If animals still worked, then they must be able to create a voltage, and from voltage comes current. If current were disallowed somehow, massive charges would build up in certain places and things would explode. A lot (assuming electrostatics still apply, which they must, because the people aren't dead).

What about this: All conductors are simply reduced in their conductivity, so things like copper and gold become the new silicon (but no one knows, because they have no functioning labs)?