I am pretty sure I know where your problem is, but bear with me as I run through the accelerator pump design and parts just so other people have it as a reference.
So in this first picture, after removing the lower bowl from the accelerator pump and getting the spring out of the way we have this view. The bowl is flipped over to the right to mirror its flange as it mates to it. The diaphragm has a seal that also seals off two small ports (arrows). The yellow arrow is the feed directly from the bottom of the main bowl through a small passage that ends in a small brass jet in the center of the accelerator pump bowl. Behind this is a check valve that prevents the fuel from back feeding into the main bowl, but also releases when the diaphragm retracts to pull in more fuel. The green arrow is the main passage that feeds up to the nozzle.
For reference purposes, here is how the plunger, diaphragm, spring, and pumper bowl are assembled in order.
Moving on the the nozzle end of the circuit at the flange side of the main bowl there is an O ring seal, a small brass jet with a spring, and a tiny ball bearing that act as a check valve to keep fuel in the nozzle ready to be fired into the throat of the carburetor. These screw in where my thumb is located and install in the order they are laid out from left to right.
This leads me to where I think your problem is. Obviously we are getting fuel into the accelerator pump bowl, so I will assume that the feed from the main bowl and that check valve are functioning properly.
I believe the issue is with the check valve on the nozzle end. If that brass jet with the spring is screwed down too far it will not allow fuel to go by, or very little, because the ball can either not move, or the spring pressure is too high for it to pump past. This jet (blue arrow) should be just about flush, or slightly below the level of, the main bowl flange. An easy way to check this is to blow through from the bottom of that passage where the accelerator pump feeds up. You should be able to blow air through relatively easily (given how small the passage is).
If you check this, adjust it if needed, and still don't get any fuel to come out of the nozzle, then my next stop would be to make sure the check valve in the accelerator pump bowl that leads to the main bowl feed is functioning properly. It is possible that this is not sealing properly - either the ball bearing is missing or maybe it is jammed into the spring - and you just are not building enough pressure and the fuel is bleeding back out into the bowl.