As a former newspaper delivery guy I can say that the distribution starts in the very wee hours. I used to pick up my Wall St. Journals around midnight. The actual delivery time will depend on it's location on the route. But delivery to all locations by 6 am is usually expected.
I figure the USA Today is printed around 1 am eastern (notice late baseball scores are rarely in the paper) locally. And ready for delivery to home office or stores a couple of hours later.