The biggest problem posed by urine, vomit, and other pet stains on leather upholstery is the lingering odor caused by bacteria. Kill the bacteria, neutralize the odor, and the leather can usually be restored.
Enzyme cleaners like Anti-Icky Poo are worth a try. Bear in mind you may have the treat the front and backside of the leather as well as internal stuffing. If enzyme cleaners don’t cut it, your best bet is a sealed ozone chamber, which will deprive the bacteria of oxygen. Fire and flood restoration companies often have ozone chambers for eliminating mold and mildew.
We have had some success with solution of 10% percent bleach in water. It will kill the bacteria, but beware, it may also damage the pH of the leather (just as acidic bodily fluids like urine or vomit can), causing it to degrade more quickly than it would otherwise. Test first in a small, inconspicuous area, allow to dry, and evaluate.
Once odor has been eliminated, any remaining stain or discoloration can be corrected with Rub ‘n Restore® Colors. However, if the leather was etched (such that it feels stiffer and harder), a silhouette of the damage will remain and the only fix is reupholstery of that panel.