Judging by previous posts, it takes a few days. The money transfer is probably automatic, the membership apparently requires some clerical input. If you don't have it in a week, then send them a support email.
https://support.chess.com/customer/portal/emails/new
Yesterday I paid for the yearly subscription of platinum membership on the iPhone app. The money (48.99$) has been withdrawn from my card (itunesshop). However, no changes to my account at chess.com have been made, I am still a regular. What went wrong? What should I do?