No, you only delete it at the end of your opponent's first turn after it's played by this effect. If another effect prevents it from being deleted at the end of your opponent's first turn when it would be deleted, that Digimon isn't deleted at the end of your opponent's following turns. If it isn't deleted by another effect when it would be deleted by this effect, it's still prevented from digivolving.

The pending processing for the effect that triggers at the end of the turn and the deletion at the end of the turn are considered to be processing that triggers simultaneously. Therefore, the turn player can choose the processing order.