A longer answer is, it depends.
It depends on whether there are reliable comms with the guns. Land lines get run over by trucks. Radios (and field phones) have batteries fail. Radios get jammed or can't reach the guns.
The guns may have other missions they are firing. They may be out of service for maintenance (only a portion of the battery at a time, resulting in fewer guns, but in 2000 many batteries are probably only one gun). They may be out of ammo. The crew may be asleep, possibly after having a little too much ethanol from the fuel tank.
But in the situation you describe, with a company commander allocated fire support and a specific mission, it is likely to be quick, especially if some target areas were pre-registered. (And pre-registering fire is routinely done when the time is available - a map and visual recon of the operational area will likely locate several target areas to have the gunners do the calculations on.)
And as for the group not being the regular FO crew, no big deal. Calling for fire is routinely taught to all officers and all combat-arms NCOs - the US Army considers it a basic military skill. A FO is a luxury that most don't expect to have.
If you want a good idea on how an artillery unit functions in combat, read Raellus's father's book
Able on the Way http://www.amazon.com/Able-Way-David...8358771&sr=1-1 . It's about an artillery unit in the Korean War, but precious little has changed between the period described there and my experience in an artillery unit in the 1990s.