I don't think it is a mistake. I suspect that this could be a US/GB difference, but I think it is fine to say 'Her son is five; he is not at school yet.'. In fact, for me, this would the more natural way to say this. If the child is too young to attend school, he is not 'at school yet'. This mother might say 'I've got three children - one at secondary school, one at primary school and one who isn't at school yet'. Speakers of US English tend to say 'in school', 'in college' and so on, whereas speakers of British English say 'at school' 'at university', and so on, to refer to a person's stage of education. There aren't many difference in preposition use between US and GB English, but this seems to be one of them.

It would be a mistake in the context of the first part. Why? The clause "he is not at school yet" means he hasn't yet arrived today at school. But why then mention the son's young age? The original sentence means that the son is too young for school.