I checked it for bugs, no errors were showing, and I've tested the mission...anything else I'm missing? Does it take time to get them published or is it automated?
When you press the Publish button it only takes a few seconds. You'll get a confirmation in your chat box and if you go to the "My Published Missions" tab it should be listed there.
If it won't publish then edit your mission, make sure the size isn't too big (bar across the bottom isn't full) and that the error "?" in the upper right isn't orange. If it is then click on it to find out what's wrong.
The only other reason I can think of that it might not publish is that you already have 3 published (i.e. it did publish and you didn't realize it) but I'm assuming that's not the case given your question.
[edit] I also think (haven't seen it myself) that if it won't publish you'll get a generic error in the chat box instead of a confirmation so keep a look out for that. See the thread "Unable to publish arc decriptions" for an example of a generic error some people are getting and possible solutions.
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When you press the Publish button it only takes a few seconds. You'll get a confirmation in your chat box and if you go to the "My Published Missions" tab it should be listed there.
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Actually, if it's the first time the mission has been published, it may take a couple minutes for the confirmation dialog to apprear (Re-publishing usually takes a few seconds). If you don't see an error message when you hit the 'Publish' button, just be patient; but you should see a confirmation message and at that point your new arc should appear in your 'My Published Arcs' tab.
Be aware that hitting the 'Publish' buttom multiple times will cause your arc to actually be publised multiple times (up to the 3 *** limit); so, if you did that, you may want to check for duplicates and unpublish those too get your slots back.
I'm sure time depends on the complexity of your arc but my 98.7% full arc didn't take even 1 minute to upload. But I can believe times would vary widely.
"My Published Arcs" - that was it. I knew "Mission" was probably wrong but couldn't remember for sure. Oh well, points for trying I hope. 
Just finished my first story/mission in AE and it won't publish it. Any thoughts as to what big issues I might be missing, but as a first time publisher am not aware of? Thanks in advance.