LCA in HLD

During the path query walk, the LCA emerges naturally. When $u$ and $v$ end up in the same chain after the while loop, the one closer to the root is the LCA. If depth[u] < depth[v], then $u$ is the LCA. Otherwise, $v$ is the LCA.

You can store this value if needed. You do not need a separate LCA algorithm like binary lifting if you only need the LCA during path queries. HLD finds it as a byproduct of the path walk.