You get paid in arrears so you should receive a payment on 25th September. I would advise you told them you are off to uni at your earliest convenience. Delaying so could show you're trying to circumvent the truth just to receive an additional payment and that is the last thing you want when you're off to uni. You have too much to lose for a few hundred pounds but rest assure you will receive your Sept payment. The date you tell them you are off to uni is the date your course starts, not the date you move to London.
Congratulations in getting into uni and I wish you the best of luck on your course.