A "furniture" dowel isn't a specific "proper" term, but generally means one of the smallish dowels about two inches long, with spiral grooves for glue to fill and adhere to, and made of some very strong type of wood. A just "dowel" can be very large or small, in either diameter, or length, or both. Furniture dowels if properly glued make a fairly strong connection, and can tolerate a good amount of stress from weight or pressure (I don't know what kind of wood they are, but I have tried to break them a few times!). Epoxy glue is not usually used for normal wood furniture repairs. However bonding into the composite materials of the steering wheel I feel it is a better choice. I use a variety of epoxy based adhesives myself, depending on the application, filling requirements, and materials being used. For something like this (a slip fit, must be able to move the dowel in place, but not be too loose), I would use the "two ton" (higher strength and slower setting time) nearly clear. I actually find that I prefer the Epoxy brand, however, I have not been able to find it locally for a couple years now. I have not tried to find it online and do not know if it is still available or not. Devcon makes a good comparable product and is available at most local hardware stores.
Finding the "two ton" version can be a bit difficult these days. Many retailers cater to the "everything instant" mentality. Several stores I used to buy epoxy from only carry the "five minute", "two minute" and even "one minute" varieties. These in the first place do not allow enough working time to get everything properly positioned. And in the second place, the final setting is not as strong as the slower setting is. A point of fact is that the fast setting glues tend to be thinner (unless you get into the "gels") and can flow deeper faster into gaps and spaces, however, any motion between pieces once setting begins results in crystallization and fracturing of the bonding. Therefore a weak set.
A couple of the steering wheel rims I have repaired (both wood and composite) have had a single break in the rim. That creates very tricky conditions for repair. One of the drilled holes needs to be deep enough to push the dowel in far enough to nearly swallow the whole dowel. With all the glues in the holes, on the dowels, the dowel has to be pushed most of the way into the deep hole. Then the rim VERY CAREFULLY sprung over to align the other hole, and the point of a small knife blade or ice pick used to slide the dowel into the opposing hole. These composite rims, especially due to age, cannot tolerate much bending, flexing or twisting of any kind. These composites reach a stress limit and with little or no warning just snap! One may fear they are not strong enough for a steering component, however once mounted onto a spider, are generally much stronger than they need to be. And always remember, anything to do with antique automobiles requires keeping one's wits about them. Things do sometime break. In the case of a steering wheel rim, IF (in this case an unlikely IF) the rim were to break? One needs to grab the spider arms and maintain control of the car. A very simple thing if one thinks about it before it happens.
For your rim, broken fairly evenly in halves. It is a simpler matter to carefully measure and drill the holes. Leave the holes just a bit longer than necessary (1/8 inch or less), and test-fit. You want it to push together easily, and align straight (how can a circle be straight????
). The holes can and really should be a bit loose (too tight and the glue will act like piston rings, seal the edges and not allow the air to escape as you push the pieces together). Use a toothpick or other small stick to apply glue all around the inside of the holes, and apply glue onto the outside of the dowels. Get both breaks ready to push together at one time. If all goes well, you may be able to just set the rim down flat on some cardboard or paper to harmlessly catch the drips. Watch for hydraulic pressure in the holes pushing the break open again! It may be necessary to put weights or clamps around the rim for an hour or so. It usually does not require much.
A small knife or sandpaper can be used to clean up and smooth the epoxy once it has dried for about ten to twelve hours. Epoxy can be added to fill surface chips or holes.
One of the tricks with most epoxy type products, is to get pretty close to a 50/50 mixture, and mix thoroughly. (Another reason I tend to dislike the fast setting types.)
That rim looks to have some nice detail. Just decoration mostly, but it would be nice to preserve it. Minor damage can often be cleaned up with knives or picks, holes and cracks filled with epoxy. Sometimes stains or shoe polishes can improve the color and blend in repairs. In some cases, a thorough cleaning followed by a flat or semigloss spray paint can make a nasty looking rim look really nice. I have been surprised in the past by how well such paint has held up with heavy use (unfortunately paint chemistry has changed so much that I have no way of knowing if current paints will do as well?)