This is all a very good question and one that I think is hard to answer.
While I think that there will be some people that will have their whole world turned upside down, I doubt it will lead to a global breakdown of law and order.
True, the majority of people on this planet are still to a greater or lesser degree religious, but there are also many people that are not. To them the notion of a god seems ridiculous. Yet these people include the same percentage of law abiding, good, clean and wholesome folks as the religious people do.
So what is keeping those people from doing whatever they want to do if they have no fear of being punished in the afterlife?
Simple, they believe in a more tangible higher power: Governments, authorities and laws.
Since according to atheists your time on Earth is limited, better act nice or you'll be wasting your time in jail.
This fear (although it's not always fear, more like a healthy respect) for authorities is instilled in us as we are children. We learn to respect and sometimes fear our parents as authorities and it grows from there.
Buddhists don't believe in a higher power, yet they are not known for going on killing sprees.* In fact, they are better known for setting themselves on fire on occasion...
*Except maybe the Shao Lin monks.