Kind of a difficult question because Tamil probably predates our earliest epigraphic records, though not by much. Our earliest dated epigraphy is from the mid second century BCE, which is about when we would expect, given the flourishing of Brahmi scripts post Aśoka. However, Tamiḷ as a language certainly predates this. Perhaps our most important literary evidence for old Tamiḷ is the Sangam literature. While dating these poems is controversial, I will err on the side of caution and say 300BCE-300CE during the classical Tamiḷ kingdoms.

With this in mind we can see that there was already a sophisticated body of work by the 3rd or 4th centuries BCE. Dravidian settlement in South India certainly predates this, going back to at least the late second millenium BCE as they slowly moved south from the Indus region. At some point, the dialects of proto-Dravidian began to split into a group of languages, with Tamiḷ being spoken furthest south. Given this, we can place the oringin of Tamiḷ as being before the 4th centiry BCE, but absolutely no earlier than the 10th and probably no earlier than the 9th. We can also reasonably date the flourishing of ancient Tamiḷ to the 3rd century BCE along with the classical kingdoms of Coḷa, Cēra, and Pandya.

The enterance into Śrī Laṅka is more controversial, no doubt complicated by ethnic conflict in the region. The Siṅhāla people arrived on an island with a fairly extensive population which they slowly pushed out. These were likely the ancestors of the Vedda people of the central highlands. The Mahāvaṁsa dates this to 543 BCE, a significant date for Buddhists, and this actually seems somewhat close. Tamiḷs are first mentioned later, during the reign of the Dravidians in the 2nd century BCE, indicating settlements were presrnt by the third, which lines up with Archeological evidence. Does this mean that Tamiḷ settlement was definitely in the 3rd century, no, it may very well predate that, but we have no evidence for that. What the evidence does indicate is a rich Tamil population in the 3rd century BCE who were very close to the now established Siṅhala people which resulted in a mixed population. Now, we can see a slight genetic and dialectical difference between Śrī Laṅkan Tamiḷs and Continental Tamiḷs that likely dates back very far due to the mixture of peoples and culture in Śrī Laṅkā.