They probably will get it, as they haven't taken their "turn" when you had your spies take a look. In Civ V, the human player always goes first on a given turn, but the AI's construction progress (on buildings, wonders, units) is not updated until the AI has the opportunity to make decisions on that turn (i.e., when it's the AI's "turn" on that "turn"). So, if you are showing 5 turns to completion and the AI is also showing 5 turns, the AI's progress will update to 4 turns when their opportunity to make decisions comes around on this turn.
You could try bribing someone to DOW Ethiopia -- that might cause them to switch away from Globe Theater to produce units.