The dates of the birth and death of Gautama Buddha are among the few things that most historians are relatively sure about when it comes to ancient Indian history. They are not fixed in stone but most historians agree about the period being 5th century BCE with the date of his death varying between 483 BCE to 420 BCE. This was near the end of the Vedic period and knowledge was still primarily orally transmitted. There is no proof that Vedic Sanskrit had a written alphabet at all.

The earliest examples of writing in Sanskrit appear from 3rd century BCE onwards during the reign of Emperor Ashoka in the Brahmi script. During Buddha's time, Sanskrit was supposed to be the language of the elite and was used for literary purposes like composing religious hymns or epics. Ordinary people of that era mostly used Prakrit and depending on the region, there were probably different dialects of Prakrit. Given that Gautama was born a prince and must have had training in elite arts, he probably did learn Sanskrit. However, he probably did learn the Magadhi variation of Prakrit too when he delivered sermons as the Buddha to ordinary people in his travels, mostly across the Magadha Empire. But there exists no written documents in India from Buddha's time and neither Sanskrit nor Prakrit seemed to have started using writing yet.

From Panini's Ashtadhyayi, a book composed of eight chapters of Sanskrit grammar, the art of writing seemed to have been known to people of that era. Panini is dated to have existed anytime between 6th and 4th century BCE but it is anybody's guess whether he wrote his book or, as was the tradition then, the book was orally transmitted. If he did write his book, he could have written it in the Kharoshti script which was used by Persian administrators in north-western regions of the Indian subcontinent. The Brahmi script that arose in India later was most likely derived from Kharoshti and was still used mainly for Magadhi Prakrit and Pali rather than Classical Sanskrit, following Panini's grammar rules, which had become the language of the elite.

TL;DR - Buddha was born in Nepal and lived and worked mostly in eastern India. As such, he was probably illiterate despite being educated as written language was still not used in most parts of the subcontinent.