Runners at 1st and 2nd with one out. Batter hits fly ball to outfield that is caught for 2nd out. Runner at 2nd tags up. Runner at 1st does not tag up. Throw to first base is overthrown in enclosed park so ball is still in play. By the time the ball is recovered and thrown to first base for the 3rd out (the runner never did tag up), the runner from 2nd base has crossed home plate. Does the run count?

Is it a force out or is it in fact a timing play?

You can never have a force out if the batter does not become a runner. In this play the batter is the second out on the fly ball. Rule 5-8b says No run can score if the last out of the inning is the result of the batter being put out before legally touching first base. In this play he is the second out. Rule 5-8b 2 says no run can score if the last out of the inning is a result of a runner being forced due to the batter becoming a runner. In this play the batter did not become a runner because he flew out so you cannot have a force out. Good rule of thumb-you can never have a force out if the batter is put out.

Rule 5-8a says one run shall score each time a runner legally touches first, second, third base and home plate or crosses the safe line (co-ed ) before the third out. A player has legally touched a base even if he left it too soon on a caught fly ball unless he is appealed for leaving it too soon. Rule 8-9h. In this play the runner from second scored before the runner from first is appealed out. As you said it is a timing issue. Is the out made before the runner crosses the plate.

In your play the run counts. You can not negate the run in your play. The only way to negate it is to have the appeal made before the runner from second crosses home plate.