For me, usually 2-3 days after the closing date. Usually, you can see how many points you're gonna get on the PDF of your bill at the AMEX Website, and in my experience for the last over 6 months, 1-2 days later it is on the HHonors website. Definitely has nothing to do with when/whether you pay the bill!

Occasionally, I have been in a situation where the HHonors Website main page shows I have the points, but when I try to book the award, the old, smaller number of points are listed. In my experience, this always updates within 24 hrs.

Overall, I would say within 3 days of your closing date, you should be able to have and use the points.