I won't give a full answer, but a way to attack the problem of computing the dimensions of the cohomology groups with compact support in the special case $n=2$.
Let $X_k = \{ (A,B) \in \mathcal{M}_m(\mathbb{K}), \ A B = 0, \ \textrm{rank}(B) = k\}$. Then, in the case $m=2$, your $X$ is the disjoint union of the $X_k$ for $k \in [0, \ldots, m]$. I will compute the number of points over $\mathbb{F}_{q}$ of $X_k$ by describing $X_k$ as a sort of iterated fibrations.
Let $p : X_k \longrightarrow \mathrm{Gr}(k,\mathbb{F}_{q})$ which sends a pair $(A,B)$ to the image of $B$. The target of $p$ has cardinality $\binom{m}{k}_q$ (where $\binom{m}{k}_q$ denotes the $q$-binomial coefficient).
The fiber of $p$ over $L$ is the variety: $$ p^{-1}(L) = \{(A,B) \in \mathcal{M}_m(\mathbb{F}_{q})^2, \mathrm{Im}(B) = L, \ L \subset \mathrm{Ker}(A) \}.$$
It easy to see that the fibers over different $L$ are all isomorphic. Furthermore, we have a natural projection $$q: p^{-1}(L) \longrightarrow \mathrm{Hom}(\mathbb{C}^m,L)_{\mathrm{rank} = k}$$ sending a pair $(A,B)$ to $B$.
Let us compute the cardinal of the target of $q$. We have a natural projection:
$$r : \mathrm{Hom}(\mathbb{C}^m, L)_{\mathrm{rank} = k} \longrightarrow \mathrm{Gr}(m-k,m)$$ which sends an element of $\mathrm{Hom}(\mathbb{C}^m, L)_{\mathrm{rank} = k}$ to its kernel. The fiber of $r$ is exactly $\mathrm{GL}_k(\mathbb{F}_q)$. Since the cardinal of $\mathrm{GL}_k(\mathbb{F}_q)$ is $\prod_{j=0}^{k} (q^k-q^j)$, we deduce that:
$$ \mathrm{card}(\mathrm{Hom}(\mathbb{C}^m, L)_{\mathrm{rank} = k}) = \binom{m}{k}_q \prod_{j=0}^{k} (q^k-q^j).$$
In order to find the cardinal $p^{-1}(L)$, we then just have to find the cardinal of $q^{-1}(B)$ for some $B \in \mathrm{Hom}(\mathbb{C}^m, L)_{\mathrm{rank} = k}$ (all such $q^{-1}(B)$ are isomorphic). But we have:
$$ q^{-1}(B) = \{A \in \mathcal{M}_{m}(\mathbb{F}_q), \ L \subset \mathrm{Ker}(A) \} = \mathrm{Hom}(\mathbb{F}_q^m/L,\mathbb{F}_q^m).$$ We deduce that the cardinal of $q^{-1}(B)$ is $q^{(m-k)m}$. This gives: $$\mathrm{Card}(p^{-1}(L) = q^{(m-k)k} \binom{m}{k}_q \prod_{j=0}^{k} (q^k-q^j).$$ This implies that: $$ \mathrm{Card}(X_k) = q^{(m-k)k} \left(\binom{m}{k}_q \right)^2 \prod_{j=0}^{k} (q^k-q^j),$$ and finally:
$$\mathrm{Card}(X) = \sum_{k=0}^{n} q^{(m-k)k} \left(\binom{m}{k}_q \right)^2 \prod_{j=0}^{k} (q^k-q^j).$$
From this formula and the Weil conjectures (and a computer), you can probably find the dimension of the cohomology group with compact support of $X$ (this is the case $n=2$).