From my experience, the cooldown seems to apply as soon as you pop the coin (not kill an ultra).
Tried popping them repeatedly until an engram dropped. It was just a waste of 30 or so coins. Went back to using one every 10 minutes or so, started dropping every 3-5 coins.
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That seems to be the consensus on here from players' experiences. So that's how I'm using them now. Thanks for your input!