Yes. Their application in the online tracker said "not started" for just about everything, while my partner's was updated to complete for almost everything.
Yesterday, a week after our oath, their online tracker (along with mine and my partners) were all updated to complete, saying that they are now a citizen.
How long has it been for your child's application since your oath?
Note that we received our certificates on Wednesday, just a few days after our oath. At that point, I think our status still hadn't updated fully. I believe that didn't happen until yesterday, though the tracker says January 11.