Whenever I take advantage of a new special offer I make a point to review the terms and conditions in detail. I often find this answers the most common questions (such as this one)

In this case, the terms and conditions pretty clearly state, "Please allow up to 8 weeks from the last day of the calendar month of your stay, for World of Hyatt Tier-Qualifying Night credits to be added to your World of Hyatt account."

In practice, it rarely takes that long (based on similar, previous offers), but that should be your expectation. Since this offer was only announced in mid to late August, and 8 weeks from the end of August would be the end of October or beginning of November (assuming the stay was actually in August), thus we could be a ways out from seeing the first nights post.

"Logistically," Hyatt tends to run these in batches and it generally will post with something like, "Reward Bonus" as the line item and then the specific number of additional elite nights.

Regards