You may expect no effect. The step controlling the electrolysis rate (i.e., the step with the lowest rate) is the diffusion of the electroactive species in the layers near the electrodes (diffusion layers).
The resistance of the solution is commonly very low (i.e., the ionic conductance of the solution is relatively high) and the solution is also stirred.
This answer is for the a common cell where the two electrodes are parallel and have the same area(i.e., uniform electric field between electrodes). For a more complex case, read something about the Hull cell.
Current is very important with electrolysis, because it is the physical number of electrons passing between the electrodes (cathode and anode) that drives the chemical processes that occur in the cell.
Faraday’s Constant is an essential tool when doing calculations related to the chemistry, rate of reactions, and other things associated with electrolysis:
Other relationships, most notably Ohm’s Law, describe the relationship between overall Power, current, and voltage:
Power in Watts = voltage X amperage.
Electrolysis cells begin passing electrons when the voltage exceeds a minimum point. From there, within a fairly narrow range, increasing the voltage does push more electrons through the cell. However, the current density also increases, and at some point, rather than driving the reaction we’re after, the higher voltage begins to cause other unwanted things to happen such as erosion of the electrodes.
In addition the overall impedance matching of the source of electricity and the electrolysis system must be considered. For maximum efficiency, the proper balance between voltage, current, and source impedance relative to the electrolysis system is very important. Strategies for doing this include adding cells in series until the sum of their voltage drops (for example, 1.5 V per cell) adds up to the source voltage.