To clarify I’m only interested in the three words update, updating and updated.
You seem under the impression that word are translatable on a one by one basis, without taking the context into account. Both assumptions are false. You often have to take the context into account to chose the right translation of a word. And the idiomatic way to express something may need to modify the structure of the sentence so that it is not possible to find the translation of a given word; that's especially true in the case of verbal forms.
As a verb form update would have to be translated as a form of "mettre à jour", which one would depend on what form you think update is. As a noun, it would be "mise à jour".
Updating is a form of the verb to update, and a difficult one to translate without context because it has no direct equivalent. "En train de mettre à jour" or "mettant à jour".
Updated is easy, assuming it is the part participle, it is "mis à jour" (but pay attention that "mis" may have to be modified to agree in gender and number with what it refers to).
See the other answers for more contextual translations, those are assuming a context and if the assumed context is the right one, they are better than what I wrote above.