This is a very basic bellows build:
Build the bellows - The bellows body can be boxes or cylinders ( round works best). Lets assume you can come up with two straight sided drywall buckets.
You make two pistons out of the lids and/or plywood. They need to slide in and out with no binding, but don't need a tight seal. You can get fancy and install a soft leather or felt rim, but I wouldn't bother.
Attach each piston to a 24" piece of 2X4 by screwing it to the center. Drill a 3/4" hole through the end of the 2X4s 4" from the end.
Make the pump lever by drilling three 3/4" holes in an 10 foot 2X4. One 4" from the end, one 34" from the end, and one 64" from the end. This will place the two buckets 60" apart. Shape the remaining end into a handle you can pump.
Make the pivot by cutting two pieces of 2X8 that are at least 30" long ( length will be determined later). Drill a 3/4" hole 4" centered from one end of each board.
Make the valves and ports - You can do all sorts of things to make the valves, but here is a simple and basic one.
Cut four pieces of 4X4" rubber or flexible leather ( heavily oiled helps) to be the valve flappers. You want the flappers flexible, but not too soft.
On the side of each bucket, around 4" up from the bottom, make a 2.5" hole. In the bottom of each bucket on the opposite side of the side hole, make a 2.5" hole through the bottom. It should be about 4" in from the side. On the inside of the bottom hole, attach a flapper along one edge to the bottom. Simplest way is to use a 4x1" strip of metal to clamp the flapper between the metal and the bottom using flat head bolts and washers/nuts .
This is the intake valve. It lifts on the draw stroke to allow air into the bucket, and seals on the exhaust stroke.
The exhaust ports are a bit trickier. First, you need to make them flat. Simplest way is to take four 6X6" piece of 2" thick wood and glue them together to make two 6X6X4 blocks. Cut one side of each block in a curve that fairly closely fits the bucket side. Simplest way is to use the bucket bottom to mark the curve and saw it on the bandsaw. No need to be perfect, just fairly close. It only needs to fit the curvature for 1" past the hole, so don't kill yourself trying to be perfect. Cut/drill a 2.5" hole in the center of the blocks. Caulk around the hole and screw the block to the bucket from the inside with flat head screws.
Now that we have a flat surface, attach the flapper on the block as you did the other ones.
Now we need a way to attach the bellows air line to the bucket. Make some sort of simple wooden box that will slip over the 6X6 block and can be screwed onto the block sides.
Since the block only needs to be big enough to mount the flapper, it might be good to find a rectangular gallon can and cut the block and flapper to fit that shape.
On the end of the box, fit a round collar that will allow clamping whatever hose you are using for the bellows lines. I would suggest 2" shop-vac hose, as it can be slipped on a PVC bushing that is mounted on the box.
That finishes the cylinders and pistons.
Make the bellows frame -
Make a frame of 2X4s, a 2X12, plywood, or anything that the bucket bottoms can screwed onto . It is just some sort of flat thing that is around 24X72". An old door would be perfect.
Use 3/4" bolts or whatever pins you choose to assemble the pump arm. One piston at the end, don't attach the pivot boards yet, the other piston at 64".
Set the buckets on the bellows frame and decide where the pivot should sit, and its length. When the handle is lifted such that the end pump is bottomed out and the other is 2" from the top, measure the length of the pivot above the frame and cut them that length.
Run a 3/4" bolt through the pivot boards and the handle arm, then screw the pivot boards in the center of the frame. This is the basic bellows. When one piston is bottomed, the other is primed and ready to pump. Brace the pivot post well with angle braces, as all the force of pumping goes to it.
Make the buckets rock. If the buckets were just screwed to the frame, the arc of the handle would make the pistons bind. The simplest was to deal with that is to allow the buckets to rock a bit as the piston goes up and down. Simplest way is to screw/bolt an old door hinge ( from the salvaged door) to each bucket. Start by bolting the hinge to the buckets so one arm is on the bucket and the other is flat on the frame ( don't attach to the frame yet). Place the hinge 90° from the ports, so the ports face sideways to the handle arm. (Make sure both exhaust ports are on the same side

). Place the hinge on the side of the buckets that is farthest from the handle.
The hinge must be elevated to allow rocking by placing a piece of 2X4 on the frame next to each bucket on the bucket side facing the handle. Screw the hinge to the 2X4 lifters first. Lift the pump handle so the far piston is bottomed ( slide a 2X4 under the other side of the bucket to make this easier to do), and then screw the lifter to the frame. Lower the handle until the near piston is bottomed and screw down that lifter the same way.
Hooking it all up -
Make some sort of "Y" connector to hook the two shop-vac hoses up to. The tail of the "Y:" goes to the air line for the forge. PVC will work fine for this. I suggest at least 24" of shop-vac hose on each bucket to allow easy of rocking. The length of hose to the forge will be determined by placement of the bellows and forge.
By pumping the handle slowly up and down you will have a continuous flow of air to the forge.
You can attach a pulley and weight to the handle so it always returns to the top position, and also can attach a linkage to a foot board to pump the bellows by foot.