What was the first language actually spoken in the Neolithic era? Into what languages did it eventually evolve?
We know almost nothing about languages in the Neolithic Era, but some paleolinguists had made many speculative deductions about the roots of Indo-European languages.
Paleolinguists is a field of study which has to depend on speculation, even where written records exist. We can take informed guesses about Sanskrit, the semetic languages of Ancient Egypt, or the language of the Olmecs. Neolithic languages predate literacy and most symbolic writing, and leave almost nothing to work with.
However, even back in the 18th Century, some linguists noticed common threads between various Indian and European languages: Agni-Ignite, Pater-Pitar, etc, and deduced that were as a common root language that influence what we now refer to as the Indo-European Languages. The linkages seem closest between Sanskrit and Lithuanian. This, however, relates to cleavages that occured in the early Bronze Age.
Speculation about Pre-Indo-European languages in Europe in the Neolithic deduce several languages, broadly grouped around what we now think of as Spain and France, Italy, the Agean, the North European Plain and the Caucausus. The only language that might remain from this time is Basque (and anyone who thinks that language is unchanged is dreaming).
The new field of Evolutionary Linguistics has even more theories, raises even more questions, and is no closer to an answer as to whether humanity has one 'mother language'. It also does not help when various nationalists feel determined to argue (without proof) that -- for instance -- Tamil or 'Proto-African' or some such thing is the world's oldest language.
We can be reasonably sure that human beings were fluently conversing with each when our species first appeared, but any attempt to understand what they were saying to each other is impossible.