I don't think they use oak. I heard they use a different wood, but I might be misremembering. I also remember hearing it has a slightly acetic character. That would be hard to get in pleasantly small doses, so maybe add vinegar to taste after it's done?
I've used some wild yeasts before, harvested from different fruits, and none of them have acted or tasted like Brett, fwiw. The closest commercial yeast in flavor to my peach-harvested wild yeast was WY2565, only the wild yeast flocced a lot better. The wild also had more undesirable fermentation byproducts (mostly higher alcohols) than the commercial one.
When people say "wild yeast" they think of crazy flavors. I'm sure there are some wild yeasts out there that do have crazy flavors, but I haven't been able to harvest any of them.