$W_\infty=\mathbb N\setminus\{1\}$. To see this, note that for $n>1$ we have a solution $(a,b)=(n,n^3)$ with $a<b$, and if we have one such solution, then $(b,n^2b-a)$ is another one (straightforward calculation) with $b<n^2b-a$, from which we easily construct an infinite sequence of distinct solutions.

For $n=1$, we just have to note that $f(a,b)=1$ iff $a^2-ab+b^2=1$ iff $(2a-b)^2+3b^2=4$ which is easily seen to have only one solution $(1,1)$ in $\mathbb N^2$, so $W_\text{fin}=\{1\}$.

Since you explicitly asked for the cardinalities of the respective sets: $|W_0|=0,|W_\text{fin}|=1,|W_\infty|=\aleph_0$.