In Pandemic, can the Coordinator's ability be used on another player's turn?
Only on your turn.

If you're asking about the first ability

All the first ability does is allow you to treat all pawns as your own when you take a move action. 

You can only move your own pawns on your turn, so you can only move their pawns on your turn. The same action cost that would apply to moving your own pawns applies when moving their pawns. 

If you're asking about the second ability

You may only take actions on your turn. You have no action points to spend outside of your turn.