How much time does it take to calculate the leveling and weighing of big and small aircraft?
Having recently weighed my plane I can tell you what's involved for determining the Basic Empty Weight for a small aircraft:


The aircraft equipment list is checked and updated.
This is kind of important as the equipment list includes the weight and arm of everything installed on the airframe. If this list is not accurate future weight and balance updates based on adding/removing equipment using the weights and arms on that list will be incorrect, and your computed weight and balance will start to diverge from the actual weight and balance until the plane is weighed again.
The aircraft interior is cleaned to "factory empty" status.
Pens, pencils, 3-year-old charts, note pads, and other things not on the equipment list are all removed, and things like tow bars are placed in their appropriate location/holder.
The aircraft is serviced with "fluids" per the manufacturer's procedures for weighing.
For most small planes this means de-fueled, full engine oil, &amp; full hydraulic/brake fluid.
The shop sets up and zeros their scales.
You tow/push the aircraft onto the scales.
(Or lift it on jack stands with integrated scales designed for this purpose.)
The aircraft is leveled on the scales.
The plane must be leveled laterally (left-to-right) and longitudinally (nose-to-tail), otherwise the computations made later will be inaccurate.
The numbers on the scales are recorded.  
The Weight and Balance is determined:


Add the numbers from all scales used. This is your Empty Weight.
You do some math on the scale numbers to determine the Center of Gravity.
(Since you know the station/arm of each scale you can figure out the CG based on the weights) 



As Terry mentioned, setting up took more time than the actual weighing and calculations. In my case I spent about a day straightening out the equipment list, and the shop that did the weighing spent about 20 minutes verifying that I hadn't missed anything. Defueling the aircraft adds a significant amount of time to the process as well. Moving the plane onto the scales and leveling it took approximately 15 minutes, and doing the math took maybe another 5 minutes. 



On larger aircraft the same general process can be used with larger scales (or jacks) - load cells integrated into a taxiway or ramp area are another option for doing this.  

I'm not sure if aircraft that can determine their operating weight (by load cells installed in the gear assembly) can also determine their own empty weight - I know the weighing report I received includes calibration information for the scales used, so presumably weighing a large aircraft using its built-in load cells would require those cells to be properly (traceably) calibrated prior to using them for this purpose.