Keep the follwing data updated during insertions: The minimum distance of a node to v a source node in the subtree rooted at v. To update you will have to update nodes on the path from the new source node to the root. For the query you also have to check the nodes on the path up to root and compare appropriately.
This will work for all but very unbalanced trees. If you still TLE you have to use for example a heavy-light-decomposition.
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Maybe you can do an “Offline Query Processing”…by taking all the inputs first and insert all the source nodes to your set and then pre-processing the distances.This is just a rough idea.
