I’m particularly interested in what those languages lacking a Christian tradition call it, since references to Adam and his apple are meaningless.
I didn’t know this in Japanese, so I had to ask my partner.
The word in Japanese is nodobotoké, which means
throat Buddha.
He also said that when people are cremated, that bone is always placed on top of the ashes. It’s held in such importance at least partly because the voice comes from there.
Roddy
The Germans have cleverly gone with Adamsapfel.
In Spanish it’s la nuez (the word also means both “walnut” and “any generic nut”); officially la nuez de Adán (Adam’s walnut), but I’ve heard the “Adam’s” part only in the context of describing men in drag. The article is important; since you’d never count Adam’s apples, saying la nuez (the walnut) automatically distinguishes it from una nuez (a nut, which would be the kind you eat).
In Catalan I’ve heard my Catalan-first-language relatives call it l’os del coll (the bone in the neck); a dictionary check tells me la nou del coll (the (wal)nut in the neck).
I checked the Elhuyar (the reference dictionary for Basque translations), since I didn’t know the word in Basque. It says zintzur-korapilo, zintzur-sagar; these amount respectively to “throat-knot” and “throat-apple”.
Bone?
In French (according to my French/English dictionary) it’s pomme d’Adam, which means Adam’s apple.
The Adam myth is shared by Christianity, Judaism, and Islam so it’s entirely possible that cultures with no ties to Christianity are calling it an Adam’s Apple.
In Polish it’s either jabłko Adama which means - wyou’ve guessed it - Adam’s apple, or grdyka.
All this etymological talk got me wondering and I found out it comes from old Polish words for choking and/or swallowing loudly. Probably shares a root with krtań, the Polish word for a larynx.
Possibly the hyoid?
Online sources indicate that the Hindi word for it is Kanth-mani, which loosely translated would mean throat-gem.
It’s always been ‘Manzana de Adán’ for me 
I realized that as soon as I hit “post,” but still- Hindi, Japanese, Mandarin, etc. all lack a Christian tradition.
I thought of that after I’d posted, but the hyoid does not form the Adam’s apple; rather it is the thyroid cartilage that forms the distinctive lump. (But yes, that is probably what is being referred to.)
In Welsh it’s afal y gwddf, “apple of the throat”. Afal Adda, “Adam’s apple,” refers to the plant black bryony. (The Breton for the body part is aval-gouzoug, which is etymologically the same as the Welsh and thus suggests the term has been around a while. They don’t seem to use “Adam’s Apple” for black bryony, though.)
Russian - кадык (“kadyk”). No relation to Adam, apples, or anything really. Sounds somewhat turkic to me, so probably borrowed from there.