I folded the ends of the new strap around the original d-rings and riveted the ends closed (just like on the original strap). I used compression rivets, but split rivets will do just as well.
Next I glued the new strap to the other internal handle parts and clamped them until the glue dried. Pretty much any glue that will stick to leather will work for this, the glue is just used to hold the parts together for re-covering. I used a silicone-based glue, but only because a tube of it was close at hand. White glue would have worked just as well for this.