It depends on the type of permit. Some permits require a few days to process and cost a few hundred bucks. Others take months (or even a year) and cost six figures. Most fall somewhere in the middle. It really depends on the type of permit you need, how closely your project follows existing rules, and how well you (or your project managers) understand the permitting process.
The best way to estimate your timeline and costs is to contact the issuing Department/Division.