Scenario: Youngster is in a swarm of three other OS ties. He has Expose as his EPT, and the other three all do that action during the activation phase. Youngster then gets one-shotted by Miranda Doni with a Torpedo.

Do the OS pilots still get to use expose during that round of combat?

The effect of expose lasts until the end of the round. They did the action before Youngster died, so they will still get the benefit.

Yes, they've already taken the action for that round. (they'd be unable to in future rounds after Youngster is gone)