What happens to DC Spindle horsepower at lower voltages?
The actual current that your spindle motor draws depends on how heavily loaded it is. If you load it down to where the power supply reaches its current limit, what happens next is determined by the design of the power supply.

If it simply goes into a constant current mode, then the spindle speed will drop. If it has a "foldback" mode or other protection mechanism, everything will pretty much just stop.

If you are powering other motors from the same supply, then when they are drawing current, the peak current that the spindle can draw will be reduced correspondingly.