Sensitivity analysis of $MST$ edges
The solution in short. For each edge $e$ not in the tree, check the path between its endpoints in the tree. The weight of each edge on this path is upper-bounded buy the weight of $e$. Keep track of the smallest upper-bound for each edge on the tree while iterating over all edges not in the tree. Following is a sketch of the correctness.

Let us call an edge in a simple cycle Heavy for this cycle, if its weight is maximum among all edges in the cycle. We claim that each edge $e$ in $G\setminus T^*$ is Cycle heavy for some cycle in $G$.

Proof. By adding the edge $e$ to the tree, it constructs a cycle with a path in the tree between its endpoints. If there is an edge in the cycle with strictly heavier weight, we can remove this edge and keep $e$ constricting a lighter tree which contradicts the assumption that the given tree is minimum.

On the other hand, it is not hard to prove that an edge is in an MST of a given graph if it is not heavy for any simple cycle containing this edge. The correctness of the sketched algorithm follows directly from this statement.