The accelerator pump has a delayer spring, clip, then a 2nd spring that goes between the delayer spring and the top. There should be a washer between the top spring and the top of the carburetor. The flat stem pump will be a bit different. There is also a WW that uses a straight round stem and may or may not have a retainer groove. The type without the groove simply sits over the pump. Using a washer at the top would also be nice, but not critical.
With the throttle off, the spring(s) are compressed, then when the throttle is pressed the springs push the pump down squirting fuel.
Hesitation when stomping on the gas pedal can be cause by:
Accelerator Pump Adjustment
Venturi Cluster - Dirty, loose screws, wrong cluster, gasket not laying down flat.
Pump Plunger Bad
Pump Inlet Check Ball - leaking, out of round
Pump Discharge Check Ball - leaking, out of round, bad seat, bad spring.
Ignition - distributor advance, etc.
WW Accelerator Pump Circuit
In this particular WW the carburetor kit will have 2 check balls. The larger one will go in the bottom of the accelerator pump well. The smaller ball goes in the bottom of the main discharge. Nothing on top of the check ball.
Fuel enters from the float bowl into the accelerator pump well via the intake checkball discussed above. The weight of the fuel and the suction of the pump moving up will draw gas into the well.
At idle, the pump is positioned at the top of the pump well waiting for acceleration.
Upon acceleration the pump moves down the well. The pressure will force the intake checkball into the hole sealing the inlet and fuel will be forced out the 2nd hole and into the main discharge.
NOTE: polish the pump well with crocus cloth to provide a smoother glide. Today's fuel has very little lubricant, which diminishes the life of the accelerator pump. Providing a smooth surface will help.
Add 2 drops of oil to the leather pump cup before installing. It may take a while for the pump cup to expand to where it will stay.
Pump Doesn't Move Down
Is the delayer spring the wrong length?
I don't know what the length of the spring should be, but when putting it on the pump you will notice it is compressed. The spring doesn't actually push the pump down but acts as a delayer cushion so when accelerating, the pump doesn't just go down all at once, but in a gliding motion. There is a return spring that goes under the pump. It is used to help return the pump and accelerator back to neutral position.
Look at this page and at the bottom is a video of how the straight shaft pump is put together.
If it's together OK, then take the top off and without the pump being in the well see if it moves up and down using the throttle. If it does, but still doesn't move down in the well, then there is something wrong on the well side, probably with the cup. Hopefully you didn't soak it in oil overnight. That will swell it up too much. Washing out with gas an letting it dry might get most of the oil out. Only use 2 drops of oil on the cup before installing. Also be sure the cup didn't get turned over when installing.