From a strictly grammatical viewpoint, both read and write and write and read are grammatical, and essentially equivalent in meaning. There is nothing inherently incorrect in referring to children learning to write and read.
Grammar is far from the only consideration in communication, however. In binomial pairs (i.e. groups of words of the same part of speech used in conjunction together), it is quite often the case that one word order is more common than the other, sometimes to the point that one is unconventional, and the same words in reverse order may not have the same meaning (hence the term irreversible binomial). There are a variety of reasons why this order can be explained, though, and pronunciation is only one.
You are correct that read and write (and reading and writing) are far more common than the reverse, but that does not necessarily mean that the use here is incorrect. Indeed, it is possible the author deliberately chose the less common order to distinguish that the children were in fact learning the specific skills of writing and reading, considered separately. Reading and writing can refer to all language skills including everything from spelling to penmanship, or by extension to all elementary school education (especially as in "the three Rs": readin', 'ritin', 'n' 'rithmetic).
Similarly, I might write that soldiers were tired and sick to specify those conditions, because sick and tired has acquired its own meaning of being exasperated or impatient. Or I might compare a duo to Garfunkel & Simon to say the "wrong" member has top billing. For a real-life example, Kraft liked to advertise its cheese and macaroni dinner, tongue-in-cheek, to emphasize how much "cheese" they packaged compared to competitors.