Improve minimum spanning tree with new edge, with better running time than O(|V|)?

Question

The problem gives a MST $T$ and a series of $Q$ queries, each one with a new edge $e = \{u,v\}$ such that no edge between $u$ and $v$ exists in $T$. For every query, we have to improve $T$ with $e$ and print the new weight of $T$.

The best I can do is run a DFS ($O(|V|)$) to find the current path $P$ between $u$ and $v$, and find the heaviest edge $e_\text{max}$ in $P$. If $w(e) > w(e_\text{max})$, we improve $T$ removing $e_\text{max}$ and inserting $e$. The overall running time for a test case is $O(Q|V|)$.

Does anybody know an asymptotically faster algorithm for this problem?