Edgar was under six months when we first saw him clean himself but we don't see it often and I can't remember the last time we saw him. He also doesn't use his cuddle buddy. Oscar was about two years old when we got him and he will hop on my arm as well as using his cuddle buddy. He often clean himself after he's done. Our newest Jackie, who turned two a couple days ago, has never cleaned himself in front of us. Jackie's only been here three months if that means anything.
With each of my chinchillas it's been different so I don't think that there is a set of rules but I'd imagine six months to ten months would be a good guess.
ETA I never noticed their penis growing but was surprised by the size when I finally saw it.