My two cents: with a modern coaxial watch (like your AT), you definitely want to go with an official Omega/Swatch service, not an independent. (I actually don’t think Donn was suggesting otherwise; I just want to be sure that wasn’t misunderstood.)
It’s true you avoid the middle-man if you do direct. FWIW, I’ve gone both routes in the past, and my AD did not add a surcharge. (One time it was warranty work, so there was no charge of any kind.)
IMO, one argument for going via the AD, though, is that it makes it easier with the return shipment. The AD will receive the watch securely and notify you when it comes in; you can go and collect it at your convenience. If you go direct to Omega/Swatch (in my experience) you will just get a shipment alert notification one day from Omega with a FedEx tracking number—and adult signature will be required. They won’t coordinate a day or time with you in advance. If you can’t be home during the day when it’s getting delivered (or if you’re out of town entirely), then that becomes an issue you need to troubleshoot with FedEx on your own. To me, as long as there’s no upcharge, the peace of mind and lack of hassle favors AD route.
There’s a thread here somewhere with current turnaround times. My recollection is that it’s about two to three months right now.
