Faster relay? 5, 12 or 24V?
We know that ampere-turns is constant for the same force, so the number of turns n must be proportional to the reciprocal of the nominal coil current. The inductance is proportional to n^2.

We can also see that coil power is generally constant for a given design of relay, across coil voltage ratings. So the coil resistance must increase with the nominal voltage squared. 

The effect of the inductance is a time constant L/R. Since L and R both increase with the square of nominal coil voltage rating, there is no first order effect on the inductance-delayed rise of force. Of course there is inertia in the moving parts and the delay due to the mechanical parts moving will not change with coil voltage rating. 

If you look at data sheets from reputable makers such as Panasonic you will see the same max/nominal operate times stated across coil voltages.  

For those who want to get an intuitive feel for this, the wire is thinner, there are more turns, and it fills about the same volume on the bobbin. 





The release time is generally stated to be lower than the operate time, but take care that this is usually specified without a flyback diode- which will greatly increase the release time.