Create Shortest Path tree for every node after Floyd Warshall in O(nm)
Compute all pair to pair distances $D[u,v]$ for $u,v \in V$ using Floyd-Warshall.
For each vertex $s \in V$ construct a graph $H_s$ as follows:

For each edge $e = (u,v)$ check whether $D[s,u] + D[u,v] = D[s,v]$. If the answer is yes the (directed) edge $(u,v)$ belongs to at least one SPT from $s$, and you add $e$ to $H_s$.

This requires time $O(n \cdot m)$ and each graph $H_s$ is encoding all SPTs from $s$. To get a single SPT from $T_s$ each $s$ compute any spanning arborescence of $H_s$ rooted in $s$. This can be done, e.g., via a BFS.