I was hoping to include more photos of the actual process, but that looks like a long way off and I don't want to lose info. Here is how Clyde Philipps explains this process
"Riveting is no big deal. Unriveting (my spelling checker doesn't like that word) can be more difficult. Let's start with that because before you need to rivet those gooseneck fittings you need to get the old rivets out. Center punch (on center!) the rivet on one head and drill out the center with a small drill to a depth just beyond the gooseneck. Its diameter should be a little bigger than the flat at the "point" of the drill that will drill out the head. This process keeps the large drill centered on the rivet (hopefully) because without this pilothole that flat point will want to walk around before it starts to drill. Though I have never used it in this process, it might be good to drill that first (small) hole, then use a countersink bit to remove the head. That way the angles will most likely be the same. But the idea here is to get rid of the head. Now, with the boom resting securely with the gooseneck fitting supported on something hard and stationary (an anvil? or the flat on a bench vise behind the jaws?) and the rivet centered over a hole or between two supports spread just enough to clear the rivet as it is driven out, go at it with a punch close to the size of the rivet and a hammer. Sometimes, in the process of riveting, the rivet shaft will swell as the head is being formed, so you may find it difficult to get the rivet moving. An old shipyard saying comes into play here: "When things don't want to move, get a bigger hammer."
Now to the riveting (and practice is not really needed; if your first head is good, it might as well be used to hold the gooseneck on the boom). Use a rivet the size of the holes in the gooseneck. On new work (a new boom) drill the wood from both sides in to the middle (that's a fun, though scary, thing to do and very rewarding when the two holes meet on center). On old work (what you will be doing) it might be a good idea to drill out the old holes in the boom a little bigger and glue in a dowel, then start from a "new work" perspective. An old hole may be oversize and will not support the new rivet when making the heads.
The length of the rivet should be the distance between the the outside surfaces of the gooseneck straight through the hole, plus three or four times the diameter of the rivet. Drive the rivet through the hole so that one end is one rivet diameter full of (beyond) the gooseneck (which will leave the other end two or three diameters full). Now, with a ball peen hammer (one that is one or two sizes smaller than the one you think is right; this requires a lot of hammering and the lighter the hammer the longer your strength will last, and lighter taps are better than heavy), and the long end on whatever you are using for an anvil, start peening over the edge of the rivet. The flat side of the hammer works well for this, but try both sides to see the difference and use the one you like best. This will start the head, increasing the diameter of the rivet end. Go around the rivet with light strokes to start the head. As the edge seems to go lower, leaving the center high, use the ball end to spread out the center. You just keep this up until the head gets bigger, the rivet gets shorter, and the countersink in the gooseneck fills with copper. You are finished with the first one with the head crowned full of the gooseneck.
Now let's see what the other end looks like. Kind of messed up, wouldn't you say? That's the reason for the extra length. It takes about a full rivet diameter on each end to make the heads, but I would not like to start out with just the needed length plus two diameters. The rivet might swell inside the boom, especially if you use heavy strokes with the hammer, and that takes from the length. After both heads are made you can file down the crown flush with the gooseneck. Now you only have two more rivets to go. These will be easier because you are now an experienced riveter.