To plate nickel out of a 1 molar Ni(NO3)2 solution should require 0.25 volts (plus a little more). But the fact that a nickel anode in H2O could generate 0.25 volts suggests that a small current could carry nickel from one electrode to the other at a voltage significantly lower than 0.25 volts. The total effect is a mechanical (physical) shift of the metal, not a change in the chemistry.
The potential for reduction of NO3- is high in 1 molar H+, but this concentration will not be nearly approached in the nickel nitrate solution (pH will probably be about 4-5). In the Nernst equation, a factor of 10000 in the hydrogen concentration and a factor of 4 in the exponent predicts that the reduction of NO3- ion will not take place at voltages which would be enough to move nickel from one electrode to the other.
Now if the current is increased by raising the voltage very high, I think that not only would the nickel be transferred faster, but some nitrate could be reduced to NO.