From Avoda Zara 57a we see that even a baby who is one day old renders the wine prohibited for drinking:
אמרו ליה רב כהנא ורב אסי לרב והא מר הוא דאמר תינוק בן יומו הוא עושה יין נסך אמר להו אימור דאמרי אנא בשתייה בהנאה מי אמרי
Rav Kahana and Rav Asi said to Rav: But wasn’t it you, Master, who said: If a gentile baby who is one day old touches wine, he renders it wine used for a libation, even though he lacks any intent? Rav said to them: Say that I said that the baby renders the wine prohibited for drinking. Did I say that it is prohibited to derive benefit from it? It is therefore permitted to sell the wine.
The same is repeated by the Be'er Hetev Y"D 124:1:
בשתיה. ואיתא בש''ס דאפילו תינוק בן יומו עושה יי''נ בשתיה וכן הוא בפוסקים
The Tur (YD 124:1) says that Only an adult, who understand what he's doing when it comes to idol worship, can make the wine asur in having pleasure from, but a child who didn't reach this definition, only makes it asur for drinking but not for pleasure.
Would this prohibition be only in the case of a gentile child who is an idolator or also in the case of a non-idolatrous one
Well, from the above sources we see, that even a child who is not an idolator, makes the wine asur to drink.