About to top 75k to make Platinum before any of my colleagues do. Before I say, 'nyah-nyah' or 'crown room, anybody?', how long does it take Delta to update someone's status in the computer, after passing a mileage threshhold?

Since I won't have my new card for a while, do I just show my boarding pass that says 'Platinum' at the door?

I'm curious because a couple years ago, there was a few weeks delay after making 50k for Gold status to show up on my printed boarding passes.

Thanks!