I'm still not convinced the question isn't trivial. Let $A$ be an associative algebra over a field $F$, and let $N$ be any nonassociative (in Serfo's particular sense of the word) $F$-algebra disjoint from $A$ with trivial nucleus. Then the Cartesian product $A\times N$ naturally inherits the structure of an $F$-algebra, and with this structure $A\times N$ is nonassociative and has nucleus (canonically?) isomorphic to $A$.

Am I missing something?

I'm still not convinced the question isn't trivial. Let $A$ be an associative algebra over a field $F$, and let $N$ be any nonassociative (in Serfo's particular sense of the word) $F$-algebra disjoint from $A$ with trivial nucleus. Then the Cartesian product $A\times N$ naturally inherits the structure of an $F$-algebra, and with this structure $A\times N$ is nonassociative and has nucleus (canonically?) isomorphic to $A$.

Am I missing something?

EDIT: This answer no longer (I think) applies to the question as it has been edited.