“One’s” is a contraction of “one is” or “one has.” It is also the possessive of “one.” Examples of these are:

One’s mood can be affected by the seasons. (possessive)
One’s the smallest positive integer. (contraction)

“Ones” is merely the plural of “one.” This is the usage you are looking for here. In English, “one” can sometimes be used to indicate individual units of something. Here are some examples. I’ve put in square brackets a translation of what “one” means in each instance:

They are the ones [the specific people] who bought the house.
I don’t play video games, except ones [non-specific games] that get left at my apartment by friends.
I am the only one [person] who can help you.
She will take in any dogs, even ones (non-specific dogs) who are not housebroken.

Note also, there are small changes you can make to such phrases that alter the meaning very slightly. With your example, there could be many variations. I think the one you want is:

I didn’t download any extra libraries except (for) the ones in the code.

The use of “for” here is optional.