boggler, I would substitute vegetable oil (safflower or similar) for the walnut oil, and then try leaving the walnuts out.
Sometimes with recipes with nuts, the nuts are part of the structure of the cake, in which case a little flour might be needed to replace the nuts. In other recipes, they are just an add-in, and no substitution is necessary. I’m not sure which way this recipe was created, so it would be an experiment to leave the nuts out without any adjustment. But that’s what I would try. Be sure not to skimp on the beating times.
You could also consider posing the question directly to Rose, over on the blog.
Good luck with all the desserts!