If you've already completed the fingerprinting for the TSA's Alien Flight Student Program and been approved to train at one flight school, how long does it take to gain approval to fly at another flight school?
Assuming that you're a category 3 applicant, you can start training as soon as they confirm receipt of your completed application but there's no specific processing time. This is from the AFSP FAQ (direct links to FAQ items don't work):
Category 3 Candidates may commence training after receiving the "Permission to Initiate Training/Fingerprint Receipt" e-mail, which AFSP issues upon receiving all required information (including fingerprints and the fee).
There is a guaranteed processing time for categories 1 and 2 (aircraft greater than 12,500lbs MTOW), however:
In accordance with Section 612 of Vision 100 and the IFR, the AFSP conducts a threat assessment on Category 1 Candidates within 30 calendar days of receiving all of the required information, including fingerprints and the fee. The AFSP conducts a threat assessment on Category 2 Candidates within 5 business days of receiving all of the required information, including fingerprints and the fee.