Through government bureacracy, and my corporate bureacracy, there is no way cell sites are getting shut down in any short time frame. It would pretty much take a cop coming in, pointing a gun telling us to shut them down before that would happen.
Posts on here stating magic switches in the EOC bunker that turn off cellsites is pure bunk.
There are no "magic switches" under the direct control of the government, but all the large US companies will honor an executive order or one from the FCC on the matter. It does take time, as you say, but it can only be a matter of minutes. By the way, one of the companies that makes the towertop and base hardware you work on is one of my clients. It's in the software, whether they tell you that or not. It can be done - whether the owner of the equipment chooses to implement it or not, and whether they tell their employees or not is another matter.
You might want to send an email to your upper management and ask them what they would do if asked by local law enforcement (after an attack) to turn the system off temporarily to prevent further detonations, and whether they would wait for a court order. The answer might surprise you.
London is not the US. In the UK, MI5 *does* have software "magic switches" and they can take the systems down *fast*. They don't have to ask the tower operators to shut them down, they can do that themselves. That came about after the IRA bombing campaigns of the 80s.