I read Jean M. Auel's prehistoric fiction series "Earth's Children" [0] when I was young. In it, Neanderthals are depicted as having a complex sign/gesture based language, with additional limited vocalisations for alert/alarm communications etc. It is a major feature of the series, and pretty interesting.
I recommend the series if you're interested in primitive skills, as it also describes many prehistoric technologies in detail (e.g. tool and weapon construction, food collection, medecine etc.). Sure, it is a work of fiction, but it's interesting and Auel did a lot of research during writing.
People get by with sign language pretty well and they typically don't use much of their vocal equipment. In fact, it's quite possible that vocal language and sign language co-evolved in earlier hominids as the vocal tract slowly evolved for more sophisticate sound generation.
I think there's study about "feral humans", kids abandoned when they were really young that managed to survive. When reintegrated into society after 0 interaction with humans they were unable to grasp more complex grammar concepts. They would get a full vocabulary but were unable to form complex phrases.
So there's definitely a component of the brain that is there but needs to be activated explicitly for "modern" humans to work as we except them to do. I don't know what the impact for sign language was though...
What if they used a click based or tone based language instead of words as we use them? These do not require advanced vocal faculties at all... We're being presumptuous here, as there have been documented instances for sapiens sapiens doing that.
Isn't it putting the cart before the horse though? Our ability to conceive of and process language developed alongside actually communicating, in an increasingly sophisticated manner. It's not like the "software" was already there, waiting for the "hardware" to catch up.
TL;DR: Analysis of a neanderthal hyoid bone indicated that neanderthals did not posses a series of air sacs, common to apes, that preclude complex vocal speech, and thus were mechanically capable of producing sounds comparable in complexity to human language.
The author Anna Goldfield also has a pretty cool podcast about archaeology and anthropology called The Dirt which I can recommend to anyone interested in stuff like this.
I recommend the series if you're interested in primitive skills, as it also describes many prehistoric technologies in detail (e.g. tool and weapon construction, food collection, medecine etc.). Sure, it is a work of fiction, but it's interesting and Auel did a lot of research during writing.
[0] - https://en.wikipedia.org/wiki/Earth%27s_Children