The phrase "nuclear family" is commonly used in English as a metaphor for the "traditional" family structure, comparing it to the structure of an atom, which "is composed of a nucleus and one or more electrons bound to the nucleus."
In that metaphor children are like electrons, bound to a nucleus comprised of "two or more adults living together and cooperating in the care and rearing of their own or adopted children". The entire phrase is what is meant to be described as a "nucleus", not merely the "two or more adults" part.
You could rewrite this sentence like so:
Two or more adults living together and cooperating in the care and rearing of their own or adopted children is the nucleus of the basic family unit.
Both this sentence and the one you quoted have the exact same meaning, all that is different is which part is presented first: the metaphor or the explanation of what a nuclear family is. Which form you use is a subjective style choice. The author of the provided quote decided to lead with the metaphor and so employed a different grammatical structure.