Here's an answer of sorts: by assumption we have that a point $((V_\cdot),x)$ of $\widetilde{\mathcal N}$ is in $\widetilde{S}$ if and only if $[f,x]=h$. So we get the obvious linearization of this condition on ${\mathfrak n}_t^*\oplus{\mathfrak n}_t$, namely that a point $(u,y)\in({\mathfrak g}/{\mathfrak b}_t)\oplus{\mathfrak n}_t$ is in the tangent space if and only if $[f,[u,x]+y]=0$.

Here's an answer of sorts: by assumption we have that a point $((V_\cdot),x)$ of $\widetilde{\mathcal N}$ is in $\widetilde{S}$ if and only if $[f,x]=h$. So we get the obvious linearization of this condition on ${\mathfrak n}_t^*\oplus{\mathfrak n}_t$, namely that a point $(u,y)\in({\mathfrak g}/{\mathfrak b}_t)\oplus{\mathfrak n}_t$ is in the tangent space if and only if $[f,[u,x]+y]=0$.

EDIT: This isn't quite right because of course this isn't well-defined for $u\in{\mathfrak g}/{\mathfrak b}_t$. I have essentially identified ${\mathfrak g}/{\mathfrak b}_t$ and ${\mathfrak n}_t^*$ with the nilradical of the negative Borel to ${\mathfrak b}_t$, but that isn't right.