Short answer is nope. That's just too far back.

Apologies in advance if you already know everything I'm about to say here. But I figured I'd cover the topic in detail for those that don't know.

Access to early language starts with writing. It's the only in-road. And writing doesn't really get underway until the middle of the 4th millennium BC, or about 6000 years ago. So that's still 8000 years after your date in question.

To understand the sound of language earlier than the earliest writing, it is necessary to reconstruct the proto language by looking for common elements in languages of the same family. For example, look at the word for tooth in languages that were derived from Latin:

diente (Spanish)
dent (French)
dente (Italian)

The sounds that are consistent with all three of these words are 'd' 'e' 'n' and 't', 'dent'. And that's the root of the Latin word for tooth (dens, dentis). So by studying related languages, it's possible to reconstruct the proto-language from which they all derived.

But that only gets you back so far. You can't extrapolate any further than that. The best studied proto-language is Proto-Indo-European, and that only gets you back to the 4th millennium BC.

As far as I know, Proto-Afroasiatic goes back the farthest, to the beginning of the 9th millennium BC. But even that is only 10,500 years ago.

So the languages of 14,000 years ago are pretty much lost. Especially for people that were living in the area of now-France back then. There's no writing in that region until the Romans introduced them to the Latin alphabet.

Basque would be your greatest bet, and it might have belonged to a larger pre-Indo-European European language family of sorts that, even though unlikely, may have dated back to that period.

Also, possible substrate influences in proto-Germanic and proto-Celtic, if true, can shed light on what languages were spoken, as some theories on early and or ancestral Germanic being a 'pidgin' or Indo-European creole of sorts and a contact language between Indo-Europeans and pre-Indo-European populations advocate.

But even then, Europe's population makeup was so entirely different and changed so radically from about ~8,000-4,000 ybp with the influx of so many populations groups, each presumably bringing their own language, languages that were likely to have been from totally different language familie(s) from those spoken in Europe 16,000 ybp.

To ask such a scholarly question and not even mention which specific region you are referring to... I don't know, man. Might get chaotic -- maybe you don't mind.
I'm not a scholar of prehistoric languages myself, but simply looking at all anthropological data as a whole gives an image of very cultured people everywhere in the world. Speech is definitely much older than 14 000 years ago. Knowledge of many hundreds of spoken words weaved into resonances of sensations, spiritual truths (life and death regarded and obscured by knowledge or lack thereof would definitely produce at least some utterances in different emotional states), as well as all the materialistic directions; warning sounds, shouts etc, even intricate/poetic stories themselves, which all can be considered primitive, yet definitely plausible for all these intelligent tribes (even just one generation "young" culturally).
This slowly creates specific language groups. These can be regarded literally as means of transportation (like genes) in the morphogenetic field of a specific organism, like a given human community, be it nomadic or stationary; evolving methods of interaction between a code -- a language -- much like the genetic code of our species evolves genetic traits. Memes are older than culture itself.
It can be stated that through being a biological and cultural part of the roots of the human race ourselves, we can access certain linguistic phenomena outside of what we've learned from direct interaction with culture. It's a form of anthropology practiced by the genes themselves. We are instantly aware of language when we are born, since that's what we all hear, unless we're impaired.
Obviously I wouldn't have been able to know any specific constructs that were (and are) being formed in my native language or any others, were I not taught basic speech by my community, but such teachings are not dependant on just that community, but all the previous ones from which they had evolved. Shamans and poets always knew there is meaning outside of the bounds of linguistic learning by the ego.
I am competent enough in speaking the Polish language (being a native speaker) to answer any questions related to the Slavic myths and constructs of language in the Slavic mythology (of language -- Sława/mowa).
Words in the Slavic language group are deeply embedded and intertwined into many current forms of language, English being one of them, throughout history. There is enough ethnological data to allow for easily explainable etymological links between Polish and words from Greek, German, Nordic and especially Vedic/Sanskrit verses and other languages, though it is hard to give specific dates reaching as far back as beyond 10 thousand years ago.
The roots of scientific observation only reach as far as some few thousand years ago -- 4 or 5. I think it's logical to assume many of the words then would have came from some primitive, Slavonic form of speech that managed to spread throughout what is now modern Poland, as well as elsewhere, since all these isolated tribes would then use the same words to describe given phenomena not connected culturally.
These cave paintings are also definitely indicative of forms of speech elsewhere in the world. We can safely say that all humans at the time possessed the ability to learn a skill such as primitive speech, but again -- no real records. Not so easy to give any dates when you can't date abstract concepts such as words, as even the highly intelligent Slavs several centuries ago did not provide any long-lasting pieces of writing -- though the word rune can be directly linked to the Slavonic word ryć -- which is one of the most astonishing, most archaic links between two very different cultural genomes; the Slavs and the Germans/Nords -- and their languages -- that is, one that can be made scientifically through etymological, geographical and cultural evaluation.