PC PSU conversion to run 12V tools
It probably detects an overcurrent and shuts off the 12V Output.

Edit: The ATX Standard states (i am guessing ist an ATX power supply) that Outputs shall be switched off if short circuit/overcurrent occures at the Output. 

That means for you: You could measure the current the drill draws, to be sure you are on the right track for solving this Problem. If there really is an OC condiction you could ramp-up the drill's Motor using PWM.
If your Multimeter does not support more than 10A just add a small current shunt resistor in series and measure the voltage over that resistor. But a Multimeter will most likely be too slow for that kind of measurement so it would be helpfull if you have access to an oscilloscope.