Can the Police or any other lawful body obtain information about a criminal from the ISP knowing such things as the shared IP (the NAT, I believe?), timestamps, visited websites etc.?
Since the ISP knows the owner (the one who pays the bills) and street address of the shared IP address police will likely start there. They might simply confiscate the systems in this home and question the inhabitants for more information, i.e. do it the analogue way and not try to somehow trace some internal computer back through the internet.