First of all, it's worth noting that the words rēx and rēs are unrelated: rēx goes back to PIE
*h₃reǵs "king" from
*h₃r-ǵ- "just" (compare rectus, rogō, rēgula), while rēs goes back to PIE
*reh₁is "goods" from
*r-h₁y- (compare Sanskrit raí "property").
Rēs is a very general-purpose word, so I wouldn't be surprised to hear it applied to speech. Its basic meaning is "thing" but it's also applied to everything from a system of government to a historical narrative to the current state of reality. In an expression like rem hanc dixit "she said this thing" it would clearly mean a statement or utterance.
I would also often translate rēs into Greek as lógos, literally "word", which is the same word used in the Gospel of John. However, in the phrase rēspublica it's definitely not referring to an act of speaking: there it has the meaning of "political system", same as in rēs nova "revolution". Rēs itself never reached Polish on its own (rēspublica was borrowed as a single unit), so if its descendant means "speech" in Polish it's a coincidence.
Rēx, on the other hand, seems to have no meanings that don't pertain to rulership. It means "king" all the way back as far as we can reconstruct into PIE, never with any meaning related to speech. King, tyrant, (metaphorically) high priest, ruling over something, master, lord, but never "one who speaks" or anything like that.
P.S. All of this is concerned with Classical Latin rather than Church Latin; in Christianity, words like verbum are almost always used instead of rēs to refer to speech. Similarly, rēx in Christianity always has the literal meaning of "king": see 1 Timothy 6:15's rēx rēgum, "king of kings". So it's safe to say the Bible had little to no impact on either of these words referring to speaking.