I did not possess a langauge until I was in Columbia School for the deaf in the 73' year which was the first 65 or so enrolled after it was built. Once I had my ABC and 123 etc in Sign the first hour the first day, being around that many deaf in a dorm situation evolved that language by the end of that first week. A month later I had pretty much all the words I needed and learned more.
You will be amused to know that the kids taught me street words and very bad language in signs that week. It was fun. Among other "Forbidden" subjects that were not taught to children back then by adults or teachers. You learn quite a lot very fast once you have a language to work with.
Then the State assigned me a voice therapist. She worked with me in spoken English for a very long time. This was from Signed English in those days to speaking english under the Galludet Signs. ASL did not exist back then. So decades later ASL was my third langauge to go with a few other extremly basic spanish, french and german for trucking purposes along with a bit of russian to order from food trucks in the NE when needed.
The Cochlear Implant creates sometimes more problems than it's worth. One of the old debating points about it is parents sometimes just install a device into the skull of a child and think good. He or she will be speaking like all normal children. Taint so. Not always. The parents never took a minute to ask the child if that child liked being deaf naturally as God made them. If you are deaf from birth and life means you don't know that you are missing out on hearing. You adapt and do well with other senses. Sometimes forcing a cochlear takes away a person's right to choice how to live or not. The parent never asked. Just spend the money and wham done. Now what?
The child in the OP's case needs to go to Deaf School in a grade appropriate and the Parents and family needs to learn sign language. The other languages will fall into place well enough. However once the child gets older its harder to learn a language.
I remember the day Johns Hopkins installed a hearing aid into me as a child. It opened up a world of hearing. However it was incredibly expensive and Father paid it all. It worked out well because i needed to hear for Trucking which was another issue into the 70-'s Deaf people did not become truckers until then. That caused other problems seperate from the Cochlear subject here.
One last thought. Do not be in the trap of discrimination for being either all deaf, a little deaf or hearing impaired as being somehow not deaf in a deaf insitution. Thats a form of racism which infected MSD for decades after I graduated under another Leader there. Some deaf or those who were hard of hearing were not treated equally as second generation deaf children of the parents attending that were my classmates in some cases.
Whatever the child has, do your best. And at some point all will be well.