Is there any way to speed up a separation packet for APFT failure?

Nowhere near as long as trying to re-enlist with a Chapter 13 RE 3

There is no set time cookie cutter format for how long it takes for that packet to be processed. I once asked this very question to the G-1 DCG of USARC and my response was anywhere from 1-18 months. It will all depend on the packet being properly formatted, all documents required to be in that packet and filled out properly, how fast it takes for each level of command (and their JAG reps) to review/approve and send up to the next level. If at any time at any level the packet is reviewed and errors are found...it is kicked back down all the way to the starting point and the process starts all over.

Whatever you do, just make sure that when you get your dd214, you have 1 and 1 for reenter code, otherwise you will no be able to comeback. Also know that sometimes Army close the doors for prior service. Last time it closed for four years, and the only way to comeback was if you have an special tab, like airborne.