There is no way to tell how long it will take. Also, if you try to call and talk to them, I wouldn't take what they tell you to be the truth. Once they recieve your materials, they will probably work on copying them and send them out with their next batch since your App is already out.
I had my profs send my stuff in by the first day, but then I got slow writing my essay, so my online app didn't get submitted till July 7th. And they STILL haven't sent my LORs out, even though my transcripts and app and DAT scores are all out.
On the bright side, I'm pretty sure that a few of the schools will look at your app before the LORs arrive b/c they realize that AADSAS is not entirely together. Good luck.