Not from experience, as it'll take me about 10 years to spend $30k on my card, but I would imagine that the award would be noted on the statement on which you crossed the threshold - and should be available to use about the same time that your miles for credit card spending post - which for me is about a week after my billing cycle closes.
Chase send award data (miles and certificates earned) to BA regularly - I wouldn't be surprised if it isn't daily so your award should be available quickly.