Help identify/source lost hardware for furniture assembly?

I took apart a single bed several years ago and have misplaced the nuts that I need to hold the side bedrails to the head and footboard.

How the bed assembles:
The side bedrails have two bolts screwed into each end. The threaded ends protrude about 2 inches from the end of the bedrail.

The threaded bolt inserts into two holes in the foot or headboard. The holes are countersunk so that when the nut is screwed onto the bolt, the head is flush with the surface of the head or footboard and is minimally visible.

The nut itself was about 1 inch long, with a flanged head, and required a hex wrench to screw it onto the bolt end. Once installed, the silver bolt was not visible; it was capped by the nut. It was a bronze or black finish. There may have been finished wooden buttons that fit into the holes, hiding the top of the nut from the eye, but I’m not concerned with finding those.

I do not want to take the bolts out of the ends of the bedrails and put in longer ones that extend past the surface of the head or footboard so a regular nut will work.

What is the name of the type of nut that I’m looking for, and where should I be able to find it?