Appeal outs are time plays, not forces. This is often a source of confusion. An appeal out does not nullify a run, unless it is the third (or fourth) out, and retires a runner that was ahead of the run.
Situation: 1st and 3rd, one out. Deep fly ball to center. The runner on first goes halfway and the runner on third tags. The center fielder looks up and sees that the runner on first has fallen down between first and second. The runner from third streaks home and scores before the ball is relayed to first to retire the runner before he can get back to the bag to tag up. Outcome: Score the run, and the side is retired on the implied appeal at first.