If we assume that 1867 represents "the late 1860s" and "aged 20, when written" refers to that same time, then the nephew in question arrived in what is now the Riverina district of New South Wales in 1850 with a father who was to be employed as what was still called an Ag Lab (agricultural labourer) in England.
So the question is "What is the possibility that a working class child would have received any formal education in a remote location where no school was built until 30 years later?" The short answer is NIL.
Of course, we might need to consider the possibility of a private tutor. The two largest "runs" in the district between 1850 and 1860 were Cuba and Tubbo. This image from the National Library of Australia shows Cuba in 1868. This is not a grand country manor which would support schooling for the families of the staff.
In the context of mid-nineteenth century rural New South Wales, it is reasonable to interpret "no education" to mean "may be able to write his own name".