Timing play or not?

[Illini_Ref]

Is this a timing play or is the check swing appeal considered the THIRD out? I guess my question is this. Is the outs considered to be made in the order they are called? For some reason a check-swing appeal seems like it is a different animal than a "normal" appeal say for a missed base.

- Runners at first and second, one out. 2-2 count
- Runners are going on a double steal
- Batter check-swings
- Catcher throws to second wildly
- Runner on second scores THEN runner from first is thrown out at third
- Defense appeals the check swing and BU rules a swing
- Does the run score?
- If it is a timing play then YES it does. If the appeal is the THIRD out then NO it doesn't