Close relative is an Exams Officer, she says AQA and all other boards will do the priority A level re-marks first in August (for those who have a Uni place depending upon it).
Then in September the Exam Boards will do non-priority re-marks which can take anything from one day to one month depending upon the subject and how inundated the subject examiners are. Most took about a week or 10 days last year.
So no rush to get your re-mark request back to school, the first week or two of term in Sept is fine - ask your Exams Office if they have set an internal deadline to receive and handle these - and you should know later in Sept what the outcome is.
Percentage of cases where the grade goes up varies but is not high, maybe 5% or less she says. Sometimes the mark goes down a bit by a few points so it's a chance you take; very unlikely for a whole grade to go down if you are near the next grade above, she has never seen this happen (but theoretically possible).