FYI, I didn't get any text regarding the credit/promo, lengthy or not.Once the phone is received and processed at the 3rd party warehouse, you'll get a lengthy text saying that the credit/promo will be applied. It's written in a very confusing way, most likely by monkeys. You won't get any other confirmation until the bill is generated.
For me it took a little less than 2 weeks to get the text. Depends on how fast the store processes their trade-in's and how fast the warehouse processes what they receive. They store likely takes a few days to process and mail the trade-in's.
(Former AT&T store manager)
I traded in my iPhone X the day after I received my 12, which was October 23, at an AT&T corp store (got receipts). A week or so later I received the return kit (for the phone that was already returned). But really nothing else.
The very next bill showed my new phone's payment, as well as the promo credit (1 of 30), as well as the sales tax. It also showed the $30 activation fee which I contested and it was immediately credited. So my only outlay for the new phone was the tax. That's actually better than what was originally promised (that it could take several billing cycles for the promo credit to appear).