I am a web developper. In my company we always use

Modifier votre statut

Modification de votre statut

Votre statut a été modifié

You can also use "mise à jour", which is a more formal translation, but less frequently used.

Mettre à jour votre statut

Mise à jour de votre statut

Votre statut a été mis à jour.

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.

To clarify I’m only interested in the three words update, updating and updated.

I imagine the context is a software with a button “Update” that changes itself in “Updating” when clicked, and then changes to “Updated” when the update process is finish.

In such a context, a correct translation would be:

update => “Mettre à jour” (beware of the accent on the a) “

updating => “En cours de mise à jour”, or if you have a moving icon close to the message, you can only use “Mise à jour” (“en cours” meaning “in progress”)

updated => “Mis à jour”

You can also use the verb “Modifier” which actually means “change” but may be correct, depending on your context.

update => “Modifier”

updating => “En cours de modification”

updated => “Modifié”