Certainly, Adam nor Eve spoke any language attributed to any culture, as there were no cultures in that time.
Grunts, groans, and hand motions would have been the only way to communicate with each other or the beasts of the earth.
GOD on the other hand, would have only planted ideas and thoughts into their heads.
Not knowing "Thought", naturally... it would have been attributed to God speaking to them.
That would be bizarre. Aside from all the telepathy and speech without words, and all the other craziness implied in your explanation, it would require that God actually planted grunts and groans in their minds, which is a huge divergence from the story.
As written, the story anthropomorphizes God. He molds clay. He breathes into it. And he speaks, endows everyone with speech, and then punishes people by giving them distinct languages.
And the LORD said, Behold, the people is one, and they have all one language; and this they begin to do: and now nothing will be restrained from them, which they have imagined to do. Go to, let us go down, and there confound their language, that they may not understand one another's speech
Of course this would have only been a temporary punishment until they became multilingual. And it would have only hurt the people living at the borders. He didn't confuse the people living within the borders of Babylon or Egypt or any other hated realm because that would have made the fable too hard to explain. But of course this is just part of the creation myth, trying to explain the origins of language before there was any knowledge of linguistics. The trick here is to make the people believe their culture and language came first, which is of course false.
I say "He" but of course the text says "us". That's another really difficult idea for the authors to explain.
Grunts and groans are impossible. Imagine trying to say "Let there be light" using grunts and groans. The only possible implementation I can thing of is Morse Code (grunt for dot, groan for dash) which takes this to new heights of absurdity. Hand signals are out of the realm of plausibility since speech is the only practical way to communicate. You can't communicate "tiger attack" with hand waving, while holding a knife and a spear and trying not to arouse the tiger.
As written -- and literally interpreted -- the Bible implies that God spoke Hebrew. Otherwise, the Bible would open as follows:
In the beginning, the Gods said: "Ugh groan groan groan ugh groan ugh ugh groan ugh ugh ugh . . " (etc.)
Also note, when the Gods said
Let there be light the void listened*, and responded with the Big Bang. A very special version of it, one that wrapped itself around the existing Heavens and Earth. Of course no one in their right mind would read this as literal truth, would they?
*even the void had ears. Everything is anthropomorphized. This is probably a remnant of the animist ideation of the early Israelite era.