No, there's a definite one-turn delay. If you're on top of it, you can remove enemy spies before you are told they are there. Similarly, if you place a spy, it cannot be removed the first turn. They will always last until at least the second turn, which means the AI suffers a similar time delay.
Theoretically, a sharp player could place their spies, remove them the next turn, place them the turn after, remove, ad nauseum and never lose a spy and keep an AIs buildings disabled half the time. You could move spies from planet to planet and get the same effect with the spies in use each turn. Moving from tile to tile doesn't work, however.