There are a few factors influencing this variation. Here's the Hebrew for the passage:
כִּ֤י לֹ֣א מִ֭מּוֹצָא וּמִֽמַּעֲרָ֑ב וְ֝לֹ֗א מִמִּדְבַּ֥ר הָרִֽים׃
The interlinear might go:
for not from-east or from-west and not from-desert lifting-up
This is preceded by a rebuke of boastful people and followed by "for only God is the judge".
Here are the difficulties:
1. This verse is highly elliptical, like much of the poetry in the Bible.
As you can see from the interlinear, there's no finite verb. Looking at the context, this also appears to be a clause, without a finite verb that could belong to it in the surrounding verses either. So the relation of "lifting up" to the other nouns is probably copular: "lifting up is not from these places".
So it would be fair to say that the NKJV is most conservative syntactically. That said, it leaves a lot to us to understand what it means for "lifting up" to not come from these places. I think the "no one" readings should probably have a thrust of "no one should look for lifting up in those places".
2. The Hebrew words for the cardinal directions are also regular nouns.
The word for "east" here is מוֹצָא motza "going out", perhaps as in the sun setting out on its journey. (Two other words for "east" are מִזְרָח mizrakh "rising" and קֶ֫דֶם qedem "earlier".)
The word for "west" here is מַעֲרָב ma'arav related to "evening", for clear reasons. (Another word for "west" is יָם yam "sea" in reference to the Mediterranean.)
Here's the part that plays into the varying translations. The third place mentioned here is מִדְבָּר midbar "desert, wilderness". But this word might also stand for "south". That south is represented by the huge desert south of Israel is also clear from נֶ֫גֶב negev "south / the Negev".
So should it be rendered "south" or "desert"? (And for that matter, "east" or "going out" and "west" or "evening"?) I think I would agree with the NIV and NLT readings here, taking east and west to represent the range of directions and then, because there seems to be no pairing of north to match south, taking that one as literally the desert or wilderness — everywhere uncharted.
Note that "lifting up" is actually a homophone of "mountains" (הָרִים harim). Perhaps this is wordplay to contrast mountains and desert, or perhaps it stands for the mountainous north (figuring in metaphors about Syria and Lebanon). Or both; this is poetry. :)
3. The central term has a range of figurative meanings.
The word translated "lifting up" is the causative participle of רום r-w-m "to rise", so "causing to rise".
NAS and BDB give a few of the various connotations this suggests:
to rise; to be exalted; to become proud; to be defiant; to remove; to be triumphant ...
There's room for interpretation. Hence the NKJV's "exaltation" but the NLT's "raise a defiant fist".
I think reading in the context of talking about the boastful and not usurping God's place as judge, the most appropriate translations will capture the idea of pride and usurpation.