The same idea of grouping terms by the number of unit bits, as well as grouping by the value of $\mathrm{wt}(j)$ (representing $j$ via individual bits) works here:
\begin{split}
s(n,m) &= \sum_{\ell=0}^n \sum_{t_1 + \dots + t_\ell \leq n-\ell} \sum_{j_1,\dots,j_\ell\in\{0,1\}} m^{\ell-\sum_{i=1}^\ell j_i} \prod_{k=1}^{\ell} (1+\sum_{i=1}^k j_i)^{t_k+1} \\
&=[x^{n+1}]\ \sum_{\ell=0}^n \sum_{j_1,\dots,j_\ell\in\{0,1\}} m^{\ell-\sum_{i=1}^\ell j_i} \prod_{k=0}^{\ell} \frac{(1+\sum_{i=1}^k j_i)x}{1-(1+\sum_{i=1}^k j_i)x} \\
&=[x^{n+1}]\ \sum_{\ell=0}^n \sum_{J=0}^\ell \sum_{j_1,\dots,j_\ell\in\{0,1\}\atop j_1+\dots+j_\ell=J} m^{\ell-J} \prod_{k=0}^{\ell} \frac{(1+\sum_{i=1}^k j_i)x}{1-(1+\sum_{i=1}^k j_i)x} \\
&=[x^{n+1}]\ \sum_{\ell=0}^n \sum_{J=0}^\ell m^{\ell-J} \sum_{d_0,d_1,\dots,d_J\geq 1\atop d_0+d_1+\dots+d_J=\ell+1}\prod_{k=1}^{J+1} \left(\frac{kx}{1-kx}\right)^{d_k} \\
&=[x^{n+1}]\ \sum_{\ell=0}^n m^{\ell+1}[y^{\ell+1}]\ \sum_{J=0}^\ell \prod_{k=1}^{J+1} \frac{kxy}{(1-kx-kxy)m} \\
&=[x^{n+1}]\ \sum_{J=0}^n \prod_{k=1}^{J+1} \frac{kx}{1-kx-kmx} \\
&=\sum_{J=0}^n (J+1)!\ [x^{n-J}]\ \prod_{k=1}^{J+1} \frac{1}{1-k(m+1)x} \\
&=\sum_{J=0}^n (J+1)!\ (m+1)^{n-J} \left\{n+1\atop J+1\right\}.
\end{split}
It also shows that the summands in the last formula corresponds to fixed values of $\mathrm{wt}(j)=J$.