Depends on several factors.
First off the VEC you used. Some are very quick to process and submit the paper work while others can be quite slow.
Next is when your paperwork reaches the FCC. They don't work weekends. If it arrives end-of-day on Friday, it's frozen until at least Monday. If it's one of their holidays (either one like ours or perhaps due to a federal shutdown or even a "closed today to save money") you'll hit a delay. As you can expect, they don't process anything on a holiday.
Next is how backed up they are. They process what they process and when they process it. They may also have a priority system where your local PD's application is processed before the amateur ones are. If they have several thousand applications arrive (say due to a narrowbanding deadline for priority agencies) it may take longer to get the amateur applications processed. If only a few applications are in the queue, they'll get them out pretty quickly.
Finally is the application on the correct form, complete, legible, contain everything they need, and the information is where it's supposed to be on the application. If there are issues, they'll need to return the application to the VEC for them to correct and resubmit. This could be where the real delays happen.
