How can I restore wood on an oxen yoke display piece?

without seeing what’s going on, I would wipe it down with an oil based stain. The wood is probably dry.

I agree, need to clean, sand and apply an oil based stain then a finish coat of polyurethane in whatever sheen you like.

I do not like to change antiques. I would clean it with a damp cloth. Rub it repeatedly until it is clean. If it is kept inside then the wood will not deteriorate. If you like to change the look you can start by just rubbing it with Danish oil or linseed oil.remember that the more you do to it the less authentic it will be.

What a catch! I would use fine sandpaper to take off whatever coating is still there. Then, I would take a jar of the lightest color wood putty I could find and go over the yoke, inch by inch, filling in all the cracks and crevices. If any of the wood seems to be cracking all the way through, I would fill both sides with putty and use clamps or very wide, strong rubber bands to hold the wood together while the putty dries. Do this to any big cracks, too. “You can never have too many clamps.” Use a clean cloth or barely damp sponge to wipe off any putty that oozes out. Let the putty dry in the yoke at least two days. Then sand the yoke, stain it your choice of wood colors, and seal it with wax glaze or polyurethane. One tip: Don’t make the yoke look too new and pretty. You probably want it to look antique and used. So add some dark stain in places before you seal it. Wagons, ho! Jewell

I would clean, sand, and move it inside.

Hope this Helps! Jeremy - https://pahjodesigns.com