Notice first that the saturation distributes over finite intersections, and for a primary monomial ideal $Q$, $\tilde{Q}=Q$ if $\sqrt{Q}\neq m$ and $\tilde{Q}=R$ if $\sqrt{Q}=m$. Thus given a primary decomposition $I=\bigcap_{i=1}^{r}Q_{i}$, $\tilde{I}=\bigcap\{Q_{i}\mid \sqrt{Q_{i}}\neq m\}$. On the other hand, $I$ is of Veronese type indexed by $d$ and $(d-1,d-1,\cdots, d-1)\in\mathbb{N}^{n}$, and we have a primary decomposition $I=m^{d}\cap (\bigcap_{i=1}^{n}(x_{1},\cdots, x_{i-1}, x_{i+1}, \cdots, x_{n}))$ by Observation 9, page 8 of the following link. So $\tilde{I}=\bigcap_{i=1}^{n}(x_{1},\cdots, x_{i-1}, x_{i+1}, \cdots, x_{n})$. Moreover, since $m^{a}:m=m^{a-1}$ for each $a\in\mathbb{N}$ (You can show first that $m^{a}:x_{i}=m^{a-1}$ for each $1\le i\le n$, and use Proposition 1.2.2 of Monomial Ideals by Herzog and Hibi), we have that $I:m^{a}=m^{d-a} \cap \tilde{I}$ for each $a\in\mathbb{N}$. Finally, since $m^{r}\supset \tilde{I}$ if and only if $r\le2$, the saturation number of $I$ is $d-2$.