The jewelry itself - as in, the metal object in itself - would probably be called טבעת if it's ring-shaped, and I'm not sure what if not ring-shared (generic תכשיט?) If you want to refer to it in the context of its usage on the body, it would be called עגיל if it's on the ear, נזם if on the nose (both ancient words), and פירסינג anywhere else. To describe the act of starting to wear ("wear"?) piercing we'd probably say the lame עושה פירסינג (but then we'd also say the lame עושה עגילים, not עונד).
To describe the act of wearing the jewel in the morning if you removed it for the night (is it done? I honestly don't know), parallel to עונד עגילים, we would probably say שם את הפירסינג. To describe the state of having piercing, also parallel to עונד עגילים, we'd probably say יש לו פירסינג.
I don't see a reason not to say עונד את הפירסינג for the last two meanings, but it sounds awkward to me - maybe the cultural dissonance between the formal, delicate word עונד and what piercing culturally stands for... and I may be over-thiking it.