Starting the core
The problem with answering this question is the way it's framed. Starting (or rather restarting) the warp core occurs in a variety of distinct stages;
- Heating up the chamber
- Allowing antimatter to enter the chamber
- Refining the mixture to allow warp plasma to be generated.
The second and third part of the sequence appears to only take a few minutes (at most) and we actually see an entire start sequence in TNG: Phantasms. The problem is that it's impossible to determine how long was spent priming the chamber in the first place.
The TNG Technical Manual describes the startup procedure in great detail but only refers to the first part of the process being done "slowly" without indicating if this is intended to take minutes or hours.
The normal power-up sequence of the engine, as managed by the MCPC,
is as follows:
From a cold condition, the total system temperature and pressure is brought up to 2,500,000K using a combination of energy inputs from the
electro plasma system (EPS) and the MRI, and a "squeeze" from the
upper magnetic constrictors.
The first minute amounts of antimatter are injected from below by the ARI. The lower MCS array squeezes the antimatter stream and
matches its aim with the MRI above, so that both streams land at
exactly the same XYZ coordinates within the M/ARC. The largest
reaction cross-section radius is 9.3 cm, the smallest 2.1 cm. The
stream cross-sections of the upper and lower MCS can vary, depending
on the power level setting.
The engine pressure is slowly brought up to 72,000 kilopascals, roughly 715 times atmospheric pressure, and the normal operating
temperature at the reaction site is 2 x 1012K. The MRI and ARI nozzles
are opened to permit more reactants to fill the vessel. The ratio is
adjusted to 10:1 for power generation. This is also the base ratio for
making Warp 1 entry. The relative proportions of matter and antimatter
change as warp factors rise until Warp 8, where the ratio becomes 1:1.
Higher warp factors require greater amounts of reactants, but no
change in ratio.
Other start-up modes are available, depending on the specifics of the
Shutting down the core
The core itself can be taken "offline" within a matter of seconds by simply mis-aligning the warp plasma conduits. This prevents the plasma from reaching the engines.
Under normal circumstances, it appears that shutting down the entire core is generally a no-no since it results in having to extensively recalibrate the chamber and start a new reaction cycle. This could, presumably, take hours to fix if it's not done properly.
It's certainly possible to start a warp core from "cold" (e.g. without priming the chamber first) so it's reasonable to assume that shuttlecraft are merely kept in a state of readiness without actually having the warp core online at all times.
TORRES: Check again. Come to join the party? We're trying to cold start the warp core for the fifteenth time. - Voy: Human Error
As far as shutdown procedures are concerned, again, we can turn to the Manual.
The normal shutdown of the WPS involves valving off the plasma to the
warp field coils, closing off the reactant injectors, and venting the
remaining gases overboard. The impulse propulsion system (IPS) would
continue providing ship power. In one shutdown scenario, the injectors
would be closed off and the plasma vented simultaneously, the system
achieving a cold condition within ten minutes. High external forces,
either from celestial objects or combat damage, will cause the
computer to perform risk assessments for "safe" overload periods
before commanding a system throttleback or shutdown.
And the system can also be shut down in a far more dramatic (albeit more damaging) way if the need arises, basically by dumping the core and
Fuel and power supplies are automatically valved off at points
upstream from the affected systems, according to computer and crew
damage control assessments. Where feasible, crews will enter damaged
areas in pressure suits to assure that damaged systems are rendered
totally inert, and perform repairs on related systems as necessary. If
the WPS is damaged in combat, crews can augment their normal pressure
suits with additional flexible multilayer armor for protection against
unpredictable energy releases. Engineering personnel may elect to
delay effecting system inerting until the ship can avoid further
danger. Exact repair actions dealing with damaged WPS hardware will
depend on the specifics of the situation.
Shutdown and Restart
This is actually rather easier to answer since we actually see it happen in Voyager: Cathexis, Tuvok shuts the engine down. Torres then needs two hours to get it back up and running:
JANEWAY: You've initiated an emergency warp core shutdown.
JANEWAY: Too late. The warp core is offline. It'll take at least two hours to regenerate the dilithium matrix. Ensign, get the rest of
the systems back online. Apparently, you've just crashed the main
computer, locked out the Bridge and stopped this ship cold. Do you
want to tell me why?