It appears from the description and the photo that the stub cross section is in essence a circle with a chord removed, basically it's a cylinder with a flat bottom. The easiest way to create a matching mortise in wood is to drill a hole of the major diameter, then partially plug it.
For a wood-only solution a small piece of dowel sanded or planed down is probably the easiest option for adding the flat. Although a face-grain plug would arguably be stronger I'm not sure it would matter, and anyway plug cutters are available in a much more limited range of sizes than dowel1.
If, as clarified in the Comments, the knobs are all going to be replaced then their shape can be much simpler, so the strength of the starting material matters less than it would if the complex form of the original knobs were being recreated. So any good, dense hardwood should suffice. Oak wouldn't be my first choice but I bet they'd outlast you. Maple would be a superior option (noticeably harder and not open-grained) but its pale colour is likely to become an issue in terms of them getting grubby with extended use. If I used maple I would consider painting them with something really tough, or alternatively staining them very dark so that if/when a clear finish wears off grubby marks are far less visible.
Consider adding a grub screw
A screw driven in through the 'bottom' of the handle at back would ensure they won't work loose over time, while still being fairly easily removable. But more than that, you could use this to avoid having to create the flat side of the mortise entirely — the screw can do all the gripping if required.
The screw length should be carefully tailored so that it pinches the stub, but does not project from the wood. Any machine screw or bolt of suitable diameter can be suitably modified with a hacksaw or junior hacksaw (to cut to length and to saw in a screw slot) and a grinder or file (to round the tip).
The hole to accept the grub screw does not need to be threaded in advance. But if you wanted to do so and don't own a tap-and-die set you can easily create an ad-hoc one, see bottom of this Answer.
Regardless of whether you thread in advance or not the walls of the holes can be strengthened by dribbling in some superglue/CA. This isn't strictly necessary
2 but by all means do it anyway if you want, it can't hurt.
1 If necessary the dowel could be custom made in the shop, the simplest method being to make a suitable dowel plate, see second half of this Answer.
2 Forced-in and cut threads in strong hardwoods are very durable, even when regularly used which they won't be here.