Since battery scientists got Nobel this week, this question is worth pondering. It is not a trivial question. However consider this question:
A car battery has 12 V, whereas 9 V cells are also common. If we connect two 1.5 V cells and one 9 V cell on series, one can generate 1.5+1.5+9 =12 V, yet this arrangement cannot start a car despite producing the same voltage.
What is so magical about the lead-acid car battery and this series arrangement? All this shows is that voltage and current produced by battery (which is the rate, hence kinetics and hence the rate of reaction) are two separate things. The main problem in batteries, the rate of electron transfer, is always a limiting factor as to how much current can be drawn from them. So far the lead acid battery is still one of the best because it has a very fast electron transfer rate unparalleled by other batteries. I do remember there was another Nobel Prize winner (Marcus?) who did a lot of work on electron transfer kinetics on the electrodes.
The Nernst potential you see for batteries is the potential when a very very small amount of current is drawn (ideally none) from the battery. This way of measurement of the potential is called null point potentiometry. Under those conditions, the rate of reaction is nearly zero because no current is being drawn from the battery.
In short one cannot measure the rate of the electrochemical reaction just from the Nernst potential because it heavily depends on the amount of current being drawn from the battery. Indeed a non-trivial problem worth someone's PhD.