In the first instance it should take a week or two. All that needs to join up is BA (GGL) correctly updating and sending the spreadsheet/update list and HHonors on the other end correctly receiving it and processing it. Plenty to go wrong there of course.
In terms of re-qualifying each Honors year a further update must be sent via GGL. Typically that best works between mid January and mid March, although its hard to tell its gone through until you are either showing Diamond on 1st April or not! Again, various things can get mixed up in these processes and by and large GGL are happy to re-look and confirm you were or were not on the update / latest list and when it is next to be sent.
There were certainly occasions in my experience of BA GGL having done the right thing and Honors dropping the ball. It proved very difficult even via Diamond line to get Honors to sort out their own mess (or even understand that BA has an agreement with Honors around GGL members and Diamond status), I usually had to await BA GGL team to re-send and cross fingers.