Minimum path - robot motion problem combined with freeze tag problem
Define a fully connected, undirected graph $G$ so that there is a vertex for the initial position of each robot, and an edge between each two vertices whose length corresponds to the time for a robot to move from one position to the other.  The edge lengths can be computed using the A* algorithm or any other pathfinding algorithm.

Now I think you want a rooted spanning tree $T$ for the graph $G$, with the following properties:


The root of the tree is the vertex that corresponds to the initial position of the awake robot.
Every node of the tree has two children, except for the root, which has one child.
The tree is a spanning tree, i.e., every vertex in $G$ appears somewhere in the three.
The "duration" of the tree is minimal among all such trees.


Here we define the duration of a tree to be the maximum of the lengths of all path from the root to some leaf.

I don't know whether there is an efficient algorithm to find such a tree.