We use these solid brass 10-32 machine screws, 1/2" long, and matching washers for fastening cast brass rivet-style bells onto leather about 1/8" thick.
How to remove rivet bells from the old strap: Remove bells from the leather strap. Cut the large round head off each old rivet (the metal "dot" you see on the backside of the old strap is the head of a rivet). A stub of the old rivet will remain in a hole that goes all the way through the base of the bell. The rivet is held in this hole by a tight friction fit.
Use a pin punch or nail set (available at hardware stores) to drive this stub through the bell into the hollow center of the bell. You can flatten the surface of the rivet with a belt sander or grinder to make this step easier. IMPORTANT: If you do decide to use a belt sander, watch the video (link is below) before you start.
Remove the rivet stub by pulling it out through one of the clean-out holes that surround the bottom of the bell. You may have to enlarge the hole slightly before the stub can be removed.
The hole where the rivet once was now needs to be tapped to accept a machine screw. Use a #10-32 tap (available at hardware stores) to cut the correct type of thread for the specific screw sold in this listing.
How to install rivet bells on a new strap: Make a new strap and punch round holes as required along the new leather strap, one hole for each bell. I use a 3/16" diameter round punch for #10 screws.
Insert a screw through a finish washer, then push the screw through one of the holes in the strap. You want the washer and head of the screw to be on the backside of the leather strap. Add a small drop of threadlocker fluid (Loctite is a common brand) or nail polish to the screw threads that are sticking out from the front side of the strap.
Thread the bell onto the screw and tighten until the base of the bell is firmly seated into the leather. Do not over tighten, or you may strip the threads in the soft brass. Video of putting bells on a strap: https://youtu.be/dAka9nQ71Es
Sold per set of screw and washer.