I think the right way is to factor $n_\lambda(q)$ in general. In particular, it is obviously a quotient of the order of $GL_n(q)$. The formula for the order of $GL_n(q)$ does not have very many prime factors: just $q$ and the first $n$ cyclotomic polynomials.
One could consider an alternate question, the order of the centralizer of $J$ in $GL(n,\mathbb F_q)$. This gives the reverse of the partial order, since the divisibility relation is reversed. The centralizer is the automorphism group of the corresponding $\mathbb F_q[x]/x^n$-module, $M$. The subgroup that fixes $M/x$ is a $q$-group, since it is unipotent. Its quotient is a product of copies of $GL(k,\mathbb F_q)$: one for each type of block, with $k$ equal to the number of that block that appears.
The number of times that the $k$th elementary cyclotomic polynomial appears in the size of the centralizer is just the sum over all sizes $n$ of the floor of $a_n/k$, where $a_n$ is the number of blocks of size $n$.
This function behaves very erratically, so we can conclude that the partial order behaves erratically as well.