It is inconsistent. A specific type of server contact has to happen for pet rescue notification to pop up. From what I’ve seen, that would be the one that happens when you enter game, the one that does a check in after you’ve been idle for a while (which can also trigger the offers popup), or the one that happens after battle. I’ve triggered a rescue on one account and then collected tribute on another in the same guild immediately after and my rescue did not appear. However, sometimes after collecting a tribute (and presumably after being afk long enough to trigger the idle-check in) you’ll get the notification and be able to enter the pet rescue. But sometimes, you’ll be able to enter the rescue and then get the notification after you’ve completed the first battle of the rescue.
This is what I would consider a bug - the wild inconsistency. Every instance of server contact (eg., checking mail, collecting tributes, opening a key) should allow you to enter an in-progress rescue, not just specific ones. If it was intended to only check post battle, it should only check post battle, but it clearly isn’t. And there should never be a situation where a pet rescue becomes enterable without generating the popup that it is available, only to have the first battle of said rescue giving you the popup.