I think the question is not about Turing completeness but about completeness in another sense: using a human language we can express a lot of formal ideas and theories given enough space.
At the same time, the animals while having complicated signal systems, cannot use their language to express complicated ideas.
It is speculated that prehistoric human languages were in this sense like the languages of animals, people could signal threats, call attention, may be express some more complicated things but their languages were too weak to express general concepts.
Similarly, people who speak different languages can communicate to a certain degree but cannot express more complicated things. And learning language in this situation may require pointing at real things or pictures when articulating, i.e. to use the means of communication outside the languages.
I encountered some claims that there are some isolated human languages which are not complete in this sense, that is it is impossible to communicate general ideas using them, but I am not sure.
That said, even well developed human languages have their limitations. They can communicate only formal ideas and cannot communicate qualia like smells, and even to communicate color the other party has to have had phsysical experience with that color. You cannot communicate what is yellow color to a creature or person who is color-blind, even if they know English very well.