That last bit stating “over time” is slightly confusing, as it has led some to assume your iOS device would notice an AirTag and alert you after 15 or 30 minutes. But
PetaPixel confirms
a report that this is not how this feature works: the privacy feature doesn’t activate a notification based on time, but instead uses location.
Your iOS device will recognize rather quickly if it senses a foreign AirTag that is traveling with you, but it won’t notify you right away. It will only notify you once you reach what is your “home.” This location of “home” is determined by your Address Book.