A current-carrying wire is not electrically charged because there are as many electrons as protons in the wire. As electrons flow in from one side, they flow out from the other, leading to no buildup of charge.
Current is the flow of charge. As long as electrons are flowing out of the wire at the same rate that they are entering the wire, no charge builds up. The wire itself is only a carrier; the charged entities are the ones that move current through the wire. All of the electrons move at the same time. When a wire attached to a light bulb is hooked up to a battery, the light bulb receives electrons from the end of the wire just as more electrons enter the battery's end. If the wire is not hooked up to a battery, the electrons do not move.
However, if one end or the other of the wire is disconnected, the wire swiftly builds up a charge. For example, if a wire is connected to a negative terminal of a battery and nothing else, the wire gains the electrical charge of that terminal, as electrons flow into it with nowhere to go. In effect, it becomes part of that terminal.