What is the voltage that goes through a computers power switch?
The current and voltage required for an ATX power supply to turn ON are tiny; connecting multiple computers GND together, and running it to another machine (your 'master' unit), however, can be problematic.   Consider some inexpensive optoisolators (4N33, under $1) instead of relays.

Wiring in parallel with the front panel switch will make both
the switch and your electronic control simultaneously effective.