Hi Rob,
First of all, what plans do you have? They should clearly show how the stock sits on the axle tree.
The stock is notched to fit over the axle housing as are the cheeks. The axle housing will be held to the bottom of the cheeks with iron straps. The cheeks will also be wrapped with iron straps which overlap the axle housing straps in the front. The trunnion cap bolts will run vertically through the cheeks and through holes in the iron straps. This will strongly tie everything together.
I would use a 1" round iron bar for an axle but this decision must be based on what your wheel hubs are. Personally, I don't think bushings or bearings are necessary, though they certainly will make the wheels roll more easily. But for the most part, you are not likely to be rolling the carriage for any sustained distance.
Find out what provision there is for an axle in the wheels you are ordering.
Hope this helps.
Zulu
Notice the notch that goes over the axle housing in this roughed out trail.
This is a split trail carriage but the theory is the same. Notice how the iron straps on the cheeks overlap the straps on the axle housing. A bolt runs vertically through the cheek to secure these straps. The top of the bolt is modified (flattened and cut for a trunnion key) so it won't pull through the straps.
Hard to tell in this picture but you can see the trunnion keys inserted into the "flattened" end of the vertical bolts.