Neglecting any resistance in the connecting wires, this situation is no different than having one large copper plate. If everything is connected together, you can't measure any voltage because the fermi levels of all the metals will be equal (at equilibrium). This is essentially shorting the terminals of a battery and the zinc will start to dissolve in the electrolyte and hydrogen will be produced at the copper electrodes. You can measure a potential difference between the copper plates and the zinc plate, if they're not shorted, but it won't be any different with one copper plate or two for an open circuit.
If you're actually using the cell to do work, you may get more work out of the cell with more copper area, as the larger surface area will result in less solution resistance and better diffusion.