Re: How long since they approve the visa?
The fastest I have received mine was in 7 days.
I was in the same situation as you once. If you need to travel before the permit arrives, go back to the Migration Office and tell them. They will check your status and see that your permit is in process and they will give you a visa for travel and re-entry. It is only for 90 days.
If you or someone who can sign for you is not there when your permit arrives by post, the post will hold it for 10 days (or something like that), and afterward, return it to the Migration Office.
When you get the Visa for travel, you tell them to make a note in your file to send your permit to your Gemeinde, who will then hold it until you arrive. You also must tell your Gemeinde you would like them to hold your card until the date you return.
It sounds complicated, but it worked just fine.