In other words, $uv = vu$ in $F_n$ if and only if $u=w^m$ and $v=w^n$ for some $w\in F_n$.
I would like to prove this without making use of Nielsen-Schreier (every subgroup of a free group is free). We can always find reduced representations $u=t_1^{\epsilon_1}\cdots t_k^{\epsilon_k}$ and $v=s_1^{\eta_1}\cdots s_l^{\eta_l}$ and the statement $uv = vu$ transforms into $$t_1^{\epsilon_1}\cdots t_k^{\epsilon_k}\cdot s_1^{\eta_1}\cdots s_l^{\eta_l}\cdot t_k^{-\epsilon_k}\cdots t_1^{-\epsilon_1}\cdot s_l^{-\eta_l}\cdots s_1^{-\eta_1}\sim 0$$ where $\sim$ denotes the equivalence relation coming from setting $x\cdot x^{-1}\sim 0$. However, the arbitrariness of $u$ and $v$ makes it hard to go on from here.