For $m, n > 1$ it's never a group.

Let $\sigma \in \mathfrak{S}_n$, $\sigma$ switches $1, m + 1$. Clearly, $\sigma \in U_{mn}$. Similarly, let $\tau$ switch $1, 2$. Then $\sigma \tau \sigma$ switches $2, m+1$. This clearly is not in $U_{mn}$.

For $m, n > 1$ it's never a group.

Let $\sigma \in \mathfrak{S}_n$, $\sigma$ switches $1, m + 1$, and in fact $\sigma = \sigma^t$. Clearly, $\sigma \in U_{mn}$. Similarly, let $\tau$ switch $1, 2$; similarly, $\tau = \tau^t$. Then $\sigma \tau \sigma$ switches $2, m+1$. This clearly is not in $U_{mn}$.