As a data point, I applied on a Sunday night with instant approval. Today I entered the club around 2pm (maybe 18 hours, and less than a business day), fully expecting them to say my membership wasn't yet active. However, I scanned my boarding pass and sure enough it accepted me. So this can be very quick.
Also, I noticed in my account that the membership expiration date is actually 15 months from when I applied, not just 12 months!