The first language of the cuneiform script was, naturally, Sumerian. The Sumerian language is related to no other language, living or dead (which makes the task of the Sumeriologists working on the Sumerian Dictionary project at the University of Pennsylvania extraordinarily difficult).
The people who rose to power in Mesopotamia after the Sumerians were the Akkadians, named for their city-state of Akkad. Their language was different from Sumerian. Akkadian was the earliest written Semitic language. There were two predominant dialects of Akkadian: Assyrian in the north and Babylonian in the south.
Sumerian was eventually replaced by Akkadian after the two languages coexisted on tablets for many years. Spoken Sumerian died out around 1800 B.C.E. Yet the language continued to exist in archaic written language, understood and read by scholars.
As a script, cuneiform was adapted by a number of groups in the Near East: the Hittites, Elamites, Eblaites (from the ancient city of Ebla), and Hurrians. Cuneiform was a script used for at least twelve different languages over a 3000-year period. Eventually, cuneiform script was supplanted by alphabetic scripts.
The last extant clay tablet written in Sumerian is dated 75 C.E. (by that time, Sumerian was a very, very old language that hadn’t been spoken for over 1800 years).