Tamil-Brahmi and Sanskrit were both derived from the Brahmi Script of the Mauryas. Tamil Brahmi evolved into Vattelattu after 200 CE. Like Tamil-Brahmi, it was used to write the Tamil language. Around 500, the Pallavas developed a new script called Grantha, or Pallava script. This was derived from Tamil-Brahmi, but was used to write Sanskrit. It was also used for a Tamil-Sanskrit hybrid called Manipravalam. This was a complicated system, using Tamil for Tamil words, Grantha for Sanskrit words. The Pallavas spread the Grantha script across South India, replacing Vatteluttu. Around this time, loanwords from Sanskrit and also significant grammatical influences were introduced into the Tamil language. This was a major factor in the shift from Old Tamil to Middle Tamil languages by 800 A.D.
The Cholas began to expand around 830. They warred with the Chalukyas and the Pallavas. The Cholas prevailed and experienced their high point in the 1100's.
They favored Shivaism, which had a strong Tamil component. The golden age of Cholas was the high point of Tamil culture and literature. Cholas commissioned a series of epics and religious literature in the Tamil language. Still, these were written in a Grantha script. The Cholas persecuted Vishnavism, which used Sanskrit, leading to a decline in the presence of that religion. The Bhakti movement produced a lot of literature; the Vishnavas in Sanskrit, the Shivaites in Tamil. There was probably a decline in Sanskrit due to the religious orientation of the Cholas, but all writing forms incorporated Sanskrit to some extent.
To the south, the Cheras and Pandyas had continued to use Vatteluttu, the wholly Tamil language. When the Cholas dominated them in the 11th century, they imposed the use of Grantha script. There is supposedly something called the Chola or Chola-Pallava script that the Cholas promoted as the defacto script in their empire. Its existence is doubted and I can't find any information about it.
I think that Buddhists would have used the Pali script.