Broken headset, repaired, but microphone doesn't work on computer, does work on phone
I think it sounds like the mic signal and headset ground lines are shorted together. The mic signal will be shorted to ground when both plugged into the same computer (i.e. shared ground)
When they don't share a ground, the problem goes away as the only common signal path is up to the short and should be of negligible resistance.
Testing with a multimeter would confirm this, just touch one probe to the tip of mic jack and other to headset sleeve (The other larger contact - I'm assuming mono TS Jacks, as opposed to TRS jacks)