What language is the music's lyrics in?

Most of the music is without lyrics. However, some tracks do include lyrics, but not all are in real languages. The ones that include lyrics in actual languages have lyrics sung in your chosen language for dialogue. (The main example I'm thinking of is "Birth of a Wish", which, depending on your language setting, will have the robots chant "This Cannot Continue" or "Kono Mama Ja Dame")

That is, if you chose Japanese audio, the lyrics will be in Japanese, whereas if you chose English, the lyrics will be in English.

The sole exception would be the "Weight of the World Medley" featured in the [E]nd of YoRHa ending, which features the Japanese, French, and English singers of the soundtrack together.

The rest of the music with lyrics, however, appears to be in a made-up language, similar to the dialogue in Gravity Rush. See this thread:

The composer said the language is based on old gaelic and what he believes Japanese could sound like in 1000 years.

So there is likely some root words and rules that are real.

Another thread discussing this can be seen here:

Its an invented language, same with the OST of NieR 1.

Follows the same route as shadow of the colossus. Mix of english, japanese, spanish, french etc.

Take a few languages like French, English, etc, try to imagine how they transform over centuries. Make some lyrics for a song.

Its based on emi evans chaos language which she created for the original Nier 7 years ago, as well as used in Drakengard 3 and obviously this game