The process Edit
First form the shoulders to the ends of the field frame bars. Below I've used a combination of hacksaw and files; a much faster option would be to use an angle grinder. Even better and faster would be to forge the shoulder, but that requires tools which I don't have.
Next make the holes to the rings. Here I've used the laborious method of drilling a round hole which I then filed rectangular. The location and alignment of the holes can be made easier by using a metal template.
Adjust the tenons and the holes as necessary to get a relatively tight fit and correct field frame alignment:
Now polish the rings. To make that easier you can make a simple wooden template which has two screws for keeping the ring in place:
For polishing I used 120 and 240 grit sandpaper wrapped around a sanding block, and finished the polishing with a steel wire brush:
At this point the ends of the tenons are still fairly long, which is problematic, especially if the field frame bars are made from spring steel or such, like here:
Therefore cut the tenons fairly short so that the heads of the rivets are easy to forge:
Forge the head using a heavy hammer first. This tightens the sides of the tenon against the hole. Then use a smaller ball-peen hammer to form the actual head: