I had the Hyatt CC up until they converted it to WOH in January.
When I had the Hyatt CC I wouldn't get my free night award until at least a few weeks after my annual fee was posted. That was consistent for the ~4+ years I've held the card. So I wouldn't expect to receive the award until a few weeks after the annual fee posts - and in any case, the T&Cs allow for a longer period (which is usually longer than it actually takes when it comes to Hyatt).