Probably no change at all.
The 'end point' for AA batteries is around 1 V, so the sharpener still ought to be producing a reasonable amount of speed and torque at this voltage. The output voltage of AAs will steadily decline from the nominal 1.5 V.
Using NiMHs will give you a more or less constant 1.2 V output. Once the cells dip below 1.1 V, there's little charge left.
If you have the NiMHs, try it and see. If you don't, try it on a lower voltage from a power supply, or three new AAs (1.6 V per cell when brand new, but falls quickly to 1.5 ish).