I plan to travel to China in May. Due to delay and uncertainty of the travel plan, I didn't submit AP application until Mar 4.
Then I realized that I might not be able to have it before leaving (someone said that it took about 4 months to get it). So I called USCIS on Apr 22 (Friday) to request to expedite the application with the reason that my hu*****and's grandma (94 years old) is in hospital and we really need to come back before she is gone. The lady who answered my phone told me that USCIS might contact me in one week to request supporting document.
I was so surprised to receive AP the next Monday, Apr 26 (without asking for any supporting document)! They must have mailed it out immediately after I called. Moreover, I noticed that the approved date is Apr 20, which is earlier than the date I called. I am not sure if this means one can always receive AP in such a period of time. But it seems that if you have waited for some time, you might be able to get AP quickly by calling them with reasonable excuse.