Abuse departments at ISPs handle a variety of issues, and corporate policies vary.
All of our complaints are handled within 24 hours by our abuse team. I happen to be the head of this team and supervise all complaints for the ISP I work for. Basically that means our technicians handle the initial contact/warning and they escalate it to me for disciplinary when the customer fails to acknowledge the complaints. I make the final decision on if it is appropriate to pursue and/or shut the service offline for them.
Although we get to complaints within 24 hours, we forward the complaint on to our clients and await their action/response.
The amount of time they are given to reply varies per the offense, but in all cases if they fail to reply to our E-Mails about the allegations within the specified period then yes, we shut their entire server down at the network level (either by unplugging the network cable, or doing it using our network utilities.)
The type of time clients are given to respond depends on the offense.
If it's SPAM, copyright infringement or something non-urgent we allow 24 hours to respond to the complaints alleged. If it's hacking or online scams its 4 hours. If it's a child pornography or otherwise illegal pornography or participating in network denial of service attacks it's 2 hours.
It's really too bad the website in question isn't hosted in our facility, or by now their server(s) could already be offline awaiting compliance to our request. ;-)
We really try to handle complaints ourself before authorities get involved, because it's always a pain when the FBI shows up with a warrant and walks away with a $10,000 server that a customer leases use of for "evidence collection". We really have to nag these agencies to let us have our hardware back, and it's hardware we could be making money off of in the mean time. If we stop it before it gets to the authorities sights, we contain the problem, keep jerks off our network, and maximize profits.