Note: I'm not a professional historian, so my knowledge of the times is a bit vague and from school textbooks.
The events of Estha and Rahel's childhood takes place in the 1960s. That is more than ten years since India gained Independence, and thirty years since the death of Sri Narayana Guru. Sri Narayana Guru had started a social reform movement in Kerala rejecting casteism and aiming for emancipation of the lower castes and casteless people. Mahatma Gandhi also strongly supported uplifting the untouchables, as did a number of other leaders of the Indian Independence movement.
The net effect was that, officially, there was no discrimination towards them, and indeed, the government had to undertake various measures for their welfare. While there was a lot of discrimination, the lower castes were not actively prohibited from education, or even prestigious jobs. Progress was slow, but there was progress. Further, Velutha's age makes it likely that his birth or childhood coincided with Sri Narayana Guru's life, so Velutha may have been a direct beneficiary of his movement.
Another factor would be the Communist movement. Kerala, the Indian state where the story takes place, was officially created in 1956, and has always alternated between Congress and Communist governments since. Kerala's first government was Communist, which started radical reforms.
Lower-caste leaders of the time, like Dr. Ambedkar, had complained about the paternalistic attitude of the upper-caste people towards those beneath, so it may also be possible that Ammu's family maybe indulging in this by "letting" the untouchable get a good job.
To conclude, the events of The God of Small Things takes place in a Kerala that was well on the way to social reform.
Personally, my father, who would be of a similar age to Estha and Rahel, and is of a similar situation to Velutha, attended the local public school, which was considered one of the best in the district, and an engineering college, and go on to get a decent job.