General Oxidant - Reductant Battery
is it true that if I connect an aqueous oxidant to an aqueous
  reductant via a wire, will a current flow along the wire? In
  particular, does current flow in the following scenario?


No true. Nothing will happen as the circuit is incomplete. The oxidant or the reductant species have to see each other "face to face". Guess what is the most powerful reducing and oxidizing agent in the world? It is the cathode or the anode respectively. Consider an electrolytic cell which is calm and unstirred. The anode and the cathode are dipping in the solution. The moment you turn on the current, only the species which is next to the electrode gets oxidized or reduced. Nothing happens to the bulk right away, so physical promixity is important in electrochemistry.