I went back and looked at the picture of your carriage, now I understand
The rounded sections, I suppose, would by necessity be the same piece of wood as the square sections, correct? This creating a static axle, where only the wheels turn. At any rate, that seems like it would be the strongest way to make it to me.
I suppose my method would probably be something along the lines of This: I would first make the ends octagonal by cutting away at the corners, to lines marked in from them to create 8 similar faces. I would then repeat the process on each of the new corners making a somewhat more circular section with 16 faces. From this point, I would probably take the drawknife and shave away at it and smooth it down to round. To ensure it is truly circular and the proper diameter, I would maybe make a simple jig consisting of 2 pieces of wood held at a certain distance apart, and then use this something like calipers to check the diameter all around.
This could all be accomplished with tools as simple as chisels, drawknives, and perhaps a good carving axe. A hand saw might be handy to make good shoulders.
this is just off the top of my head how I would do it, there may be a better way but this makes sense to me.