Mrs. Mann teaches Oliver the basics of reading, but not how to write.
When Oliver Twist was born, his mother was an unwed mother with few options. She ended up dying in childbirth, and he was left at the mercy of the system. Unfortunately, that meant he ended up in a workhouse run by Mrs. Mann. In this place he was educated, although he was barely educated. The title of Chapter 2 is mostly sarcastic. The children are mostly neglected at Mrs. Mann’s house.
[They] rolled about the floor all day, without the inconvenience of too much food or too much clothing, under the parental superintendence of an elderly female, who received the culprits at and for the consideration of sevenpence-halfpenny per small head per week. (Ch. 2)
Somehow, he does learn to read while at Mrs. Mann’s house though. We know this because when he arrives at Fagin’s he is able to read. Fagin gives him a book about criminals and he is able to read it.
Here, too, he read of men who, lying in their beds at dead of night, had been tempted (so they said) and led on, by their own bad thoughts, to such dreadful bloodshed as it made the flesh creep, and the limbs quail, to think of. (Ch. 20)
Later, he tells Brownlow that he would like to read his books. So somewhere along the way, Oliver learned to read. However, he does not seem to be able to read well. This makes perfect sense. I doubt Mrs. Mann, who was very neglectful, spent much time teaching him. He might have been self-taught at best.
Oliver finally gets a better education once he goes to Mr. Brownlow. It is specifically indicated that Brownlow was the first person responsible for improving not only his circumstances, but also his education.
Every morning he went to a white-headed old gentleman, who lived near the little church: who taught him to read better, and to write: and who spoke so kindly, and took such pains, that Oliver could never try enough to please him. (Ch. 32)
Therefore as soon as Oliver gets away from the criminals and with his real family, and his proper social station with Brownlow, he finally learns to read. He also learns to write, which shows that if he did know how to read before he did not know how to read well.
Oliver was neglected for most of his childhood. He goes from the workhouse to a failed apprenticeship as an undertaker, which he runs away from. Fagin takes him in and tries to tempt him to a life of crime, until he finally finds a family with Brownlow. Dickens is trying to tell us that Oliver remains innocent and good despite a lifetime of potential corruption. The fact that Oliver might know how to read is an idealistic oversight perhaps, but it moves the plot along and allows for further potential temptation of Oliver. Dickens corrects it by allowing Brownlow to improve Oliver’s education.