Eric, that is really a cyclical thing your asking. I waited 3 weeks, some friends of mine only waited 1 week. Some people I have talked to waited 2 months or longer. The Aviation pipeline over the last year has really gone through an overhaul, as the Navy is striving to do away with the pools.
But alot of that also has to depend on what time of year it is also, during the winter months the three squadrons at Whiting drop some of their student loading due to inclement weather, and not having the same number of flying days they have in the summer. But the way things have been shaping up lately, I would imagine that you should expect less then a 3 week wait.
If they send you to Corpus Christi, expect a little delay for checking out and into a new command, along with the moving time. Trust me, after API, you WILL want a little time off, but right now I am on my 5th week of waiting to start Advanced (PROPS) here in Corpus, and still have 3 weeks to go. While it has been nice to have time off, I am itchin to get back in the plane, albeit I will have to go through another Ground School first! It is something you will definitely get a feel for as you go through API, and hear first hand what your wait in the pools is going to be. Hope that answers the question.