Let's say, for whatever reason, someone found a lost object but did not get around to publicizing it for over a year. Does the Mitzvah to publicize it still apply, assuming that after the long duration the owner is probably yeush (has abandoned hope)?
In short, yes.
You seem to be talking of a case where someone found a lost object that had identifying marks and some degree of value, and there was no reason to believe that the owner gave up hope of finding it before you picked it up. That would give the finder the requirement to guard it which, at that point, can't be nullified by the owner giving up hope. The finder must still publicize it and return it.
See this article by Rabbi Neustadt who reviews the relevant halachos (he cites the original sources there). The relevant section is:
So if someone found the item before the owner gave up hope, they are still obligated to return it.