Here's a data point for all interested:
I signed up for the $200 Gift Card offer on the Disney Rewards Card (NO Annual Fee Version)
Applied: 8/7/17
Met $500 Spend: 8/16/17
Statement Closed: 9/7/17
Gift Card Received 9/29/17
There was a letter with the gift card dated 8/21/17...not sure how it took 5 weeks to mail this letter, but there you go folks!
Gift Card was received 44 days (just over 6 weeks) after meeting the minimum spend, Roughly 3 weeks after my statement closed.