It supposed to be the next day. It didn't work out that way for me. I called AMCAS a week after my letters were assigned to them in VE. They told me to call my advisor. I did, and he re-assigned the letters to AMCAS through VE and they downloaded them the next day.
If it's been awhile, I'd contact your advisor. They can see which schools (or AMCAS) have accessed them. If AMCAS hasn't even looked at them yet, chances are getting him/her to re-assign them would help.