Pretty much all the "nutrients" (I'm assuming you mean vitamins here) are going to be inactivated by the soaping process. It doesn't matter whether it's CP or HP. Some things like the metals (ex: magnesium, calcium, iron) survive the process but, you don't want those in soap anyway. Plus, they're going to be bound up by the soap. Vitamin E also apparently survives saponification. Rosemary oleoresin will also survive the process.
Things like vitamin A and C will not survive. In fact, they are very sensitive and require certain conditions to retain their activity. Even if they did survive the initial process, they would be inactivated very soon into cure.
Adding extra fats in at trace does not give that fat as the sole superfat. Saponification has just begun at that point so adding extra in at that point will do nothing more than make it more likely to forget it! You can control the SF in HP though.
Adding things after the cook likely won't make a difference to nutirent survival. You're still looking at an alkaline environment and that will make most vitamins or other antioxidants degrade. Soap is not a good substrate to preserve nutrient quality. Some components in EOs will also change during saponification and form different compounds. Soap making is a very harsh process and it is difficult to say a soap will do anything but clean.