E.S.
There are different theories on stuttering - that it's 100% physical (a neurological problem between the brain's commands and the mouth's movement) or that it has psychological roots based on trauma or anxiety. Being anxious yourself, impatient, critical, etc. will only add to the boy's problems, and like the previous woman, may be a trigger for your daughter - something bad she can do to "get" you on occasion as she's mimicking the boy. If the mom that you're the nanny for is sensitive about how her kids act, this could contribute to her boy's lack of confidence in speaking. Drawing attention to it yourself is unlikely to help. More likely, the most help you could be is to remain calm and patient while the boy is trying to speak, same thing with your daughter. Virtually everyone has stuttered at one time or another... Good luck, practice deep breathing and patient smiles!