I restored a steering wheel. Done a couple of them actually. These restorations were "partial" and did not involve making the grooves the entire circumference.
I simply matched the new groove (cut in epoxy) to mate with the adjacent grooves in the original material. I can't say I measured it, but I think about 1/16" or maybe slightly deeper. 1/8" would be too deep.
I made a tool starting with a tapered wooden dowel to fit the center taper hole of the wheel hub. Then a pivot made of a No. 12 wood screw (maybe 3/16" diameter?) and then a wooden pivot arm long enough to pass over the wheel outer circle and such that it could be swung on the pivot.
To this I attached a wooden block with (first) a piece of HHS from a lathe tool holder ground to the correct thickness. This worked pretty well at first, but I found the tool dulled quickly in the bakelite material and had to be re-ground quite quickly. Then (second) I discovered a hacksaw blade across the tooth kerf was just about the same thickness as the groove. This I substituted for the HHS cutter and the multiple teeth lasted much longer. And seemed to cut better and without getting hung up in the groove as much.
One caution though - the wheel exterior hub is NOT round. Well not strictly. It varies a bit like off center - and also is a bit egg shaped. In my case it mostly made no real difference as I was only trying to match a short segment of broken out/repaired covering. Only once did I have to change the centerpoint of the pivot slightly by remaking the center dowel a little "off center" - and this to compensate for "entry" and "exit" alignment of the new groove in what I think was the largest length repair.
One repair segment did not involve all the four grooves (four? IIRC) Instead I had the additional challenge of making the grooves "parallel." You may likely have fun like this too.
Anyway, I found afterwards that during hand sanding with 200 grit and leveling the wheel covering, the epoxy picked up black from the original bakelite and really couldn't be told color wise from the rest of the wheel. Still I ended up painting the wheel with spray lacquer just to levelize and obscure any transitions in the repair between new and old material. It came out looking quite original actually.
I hope it stays that way. Some people have complained of black wheel coatings coming off on their hands while they drive.
One more warning: as you pick out a used wheel for this sort of repair, try to choose a wheel without loose "thingies" (rust chips?) floating around inside the tubular rim. The second wheel I considered only slightly less successful when I discovered after repair that "something" is floating around inside. (!)
One may not notice it on a wheel mounted on a steering column.
Good luck with yours. For the investment of a $3 broken wheel (I love Amherst, NH antique auto flea-market!), a $2 hacksaw blade, a $5 Siamese tube of epoxy, and a $6 can of lacquer, it made a pretty nice looking wheel.
Joe K