I've been on the phone with various departments of AT&T (and Apple) all day. Literally...about 6 hours today.
First of all, do not believe any support rep who tells you this is Apple's fault. It is not. The fault lies squarely at the feet of AT&T.
I got up high enough in AT&T's iPhone Activation Center hierarchy to be told, "We weren't prepared. The system is automated, but it is overloaded. We have had staff (at the aforementioned Indian call center) helping out with manual processing of activations."
At about 18 hours now, I've been told to expect activation within 7 more hours. They put an extra flag on my account to expediate processing, but I don't believe it will do any good. And only at the point of activation can the porting process from Verizon even begin. So don't believe any support rep who tells you there is a delay because you're porting your number. They don't know what they're talking about. Activation is completely separate process, and entirely internal to AT&T.
Anecdotally, when I called the Verizon port center today, a very peppery woman told me that AT&T has been experiencing outages in their port transfer system for a couple weeks now. Her tone did not appear scripted, but instead the result of many dealings with them.
I've been buying/using Apple products since 1986. This has been the worst "Apple Experience" I've ever had. I suggest that everyone involved in this fiasco let Apple know what you think of your experience. And that you ask for your pound of flesh in return for the inconvenience. Apple knows that AT&T not only dropped the ball, but kicked it into the sewer, and so they will more than likely go to bat for you.