great uncle or aunt
You can adopt someone to be your child but to be your nephew he had to be adopted by your siblings. Then he will be your nephew. There is no nephew-adoption.
you would be the child's great aunty or uncle Therefore, they would be your great niece or nephew.
Your child and your nephew are first cousins. Your child and your great nephew are first cousins, once removed.
your sister's child is your nephew
Your son's child and your nephew are second cousins.
Your great nephew's child is your great great niece (if a girl) or great great nephew (if a boy).
Your nephew's child is your great niece (if a girl) or your great nephew (if a boy).
Your great great nephew's child is your great great great niece or great great great nephew, depending on the sex of the child.
Your brother's or sister's son is your nephew. Your nephew's or niece's child is your grand-nephew or grand-niece.
The child of your nephew is your great nephew (if a son) or your great niece (if a daughter).
Probably none as I dont think I know you. If you and I are siblings then my child, if a boy, would be your nephew.
If there are no other family members, and you are no longer able to care for this nephew, the only remaining alternative is for him to become a ward of the state. Contact the child welfare agency in your state.