With no runners on bases, the batter hits a ball to the short-stop. The batter steps over first base without making contact with it. The first baseman drops the ball and never had possession. Knowing that the runner didn't touch first base, before the runner returns to the first base the first baseman picks up the ball and touches the base. What's the call?
Until either the runner crosses first base, or a force out or tag has been applied, the umpire will not make a safe or out call.
Given this scenario, the umpire will not make a call either when the runner oversteps the base, or when the ball is dropped by the first baseman. As you state, the next action is the tagging of the base by the first baseman with the ball, and so this would result in an out.
The same can be seen in close plays at the plate. Often home plate is overslid or missed, and a tag is also missed. The umpire will watch intently until either the runner is tagged, or home plate is touched.
This is best covered by MLB rule 5.09(c)(2)
5.09(c) Appeal plays
Any runner shall be called out, on appeal, when:
(2) With the ball in play, while advancing or returning to a base, he fails to touch each base in order before he, or a missed base, is tagged;
The ball is still in play, the runner has failed to touch first and has not yet returned to the base, and the fielder tags the missed base. The runner is out.
This rule does not address the OP's scenario. Apr 7, 2021 at 10:57
This is not an appeal play necessarily- if the ball is still in play and the runner has not advanced to second. If the runner just overran the bag it’s just a normal putout.– Joe ♦Apr 7, 2021 at 14:24