When will it post
 

I'm already qualified as FO for 2015.

My current MDM is about 37k, and in the next couple of weeks, my Amex spend will hit $25k. When I hit the $25k Amex spend, I'll get an additional 10k MQMs. On top of that, I have a TATL trip coming up that should push me over the 50k MQM mark.

From that point, how long will it take for the DL system to recognize that I've qualified as a GM?

The reason I ask is that five days later, I'm flying VIE-AMS-BOM on a cheap T fare. I'd love to be able to take advantage of the KLM lounges in both VIE and AMS on my return, but that will only happen if DL updates its systems to reflect my recently earned status.

Is this reasonable?

Amex bonus MQM's won't post until after your billing statement closes (just like the other miles you earn from Amex), so it really just depends on the timing of your Amex billing cycle.

Using partner lounges often requires the actual GM card, but maybe they will accept the card from the FlyDelta app.

Also keep in mind the May 1 end date for non-partner airline lounge access when flying internationally. You won't be affected at AMS since it is a KL partner lounge, but it could affect your future travel plans.