You can preprocess your tree in time $O(n \log n)$ so to obtain a data that uses $O(n)$ space that can answer path-minimum queries in constant time. See "Bottleneck Edge Queries" in this paper which shows how to answer queries that ask for the edge of minimum weight in the unique path between two given vertices in a tree.
You can easily reduce your problem by splitting each edge into two edges and then setting the weight of each edge incident to a vertex $v$ of the original tree to the weight of $w$ itself.
If you are find with a solution that is easier to implement but requires $O(n \log n)$ space and preprocessing time (rather than the $O(n)$ space of the paper I liked above), and supports $O(\log n)$-time queries, then you can do the following:
Start by constructing a LCA oracle for $T$. To do so consider an Euler tour $E = \langle u_1, u_2, \dots u_{m}\rangle$ of your tree and construct an array $A[1 \dots m]$ where $A[i]$ contains the depth of $u_i$. Here $m=2m-2$. Then construct a range minimum query (RMQ) oracle on $A$, so that the LCA $u_k$ between $u_i$ and $v_j$ with $i<j$ is such that $k$ is the index of the minimum $A[k]$ in $A[i], \dots, A[j]$.
A simple RMQ oracle is the following: for each index $i$ of $A$, and for each $j=0, 1, \dots, \lfloor \log m \rfloor$, store in $B[i, j]$ the index $k$ of the minimum $A[k]$ between $A[i], \dots, A[i+2^j-1]$. All the entries of $B$ can be found in $O(n \log n)$ time since, for $j>0$, $B[i,j]$ is either $B[i, j-1]$ or $B[i + 2^{j-1}, j-1]$.
The answer to a RMQ query $(i,j)$ is then the index of $A$ with the minimum value among the two indices $B[i, \Delta]$ and $B[j- \Delta, \Delta]$, where $\Delta=\lfloor j-i+1 \rfloor$.
We are now able to find the LCA $w$ of two vertices $u$ and $v$ in constant time. Then your problem reduces to the one of finding the vertex of minimum weight in two ancestor-descendant paths, namely those between (i) $w$ and $u$, and (ii) $w$ and $v$.
This can be done using a trick similar to the previous one: For each vertex $z$ at depth $d$ maintain $\delta = \lfloor \log d \rfloor$ values $M[z, 0], \dots, M[z, \delta]$, where $M[z, j]$ is the minimum among the weights of the vertices $z= z_1, z_2, \dots, z_{2^j}$ encountered by walking from $v$ towards the root for $2^j - 1$ steps. Moreover, for each vertex $v$, we explicitly store the $O(\log n)$ vertices $z_{2^j}$.
As before, all the values $M[z,j]$ can be found in $O(n \log n)$ time and a path query on an ancestor-descendant path of length $L \ge 2$ can be answered by letting $\ell = \lfloor \log L \lfloor$ and selecting the minimum among $M[z, \ell]$ and a recursive query for the path of length $L - 2^\ell$ that ends with $z_{2^\ell}$.
The query time can actually be improved to $O(1)$ if you additionally compute a long-path decomposition of your tree. The long path decomposition is defined recursively and is obtained by selecting the longest root-to-leaf path $P$, deleting (the edges and vertices of) $P$ from the tree and returning the union of $P$ with all the long-path decompositions of the trees in the resulting forest.
This decomposition can be found in $O(n)$ time and has the following property: given a vertex $v$, the (unique) path $P_v$ that contains $v$ has a lenght |P_v| not smaller than the height of the subtree rooted in $v$.
Let $P_1, P_2, \dots$ be the paths of the decomposition and for each $P_i$ construct an array $P'_i$ by extending $P_i$ by $|P_i|$ edges towards the root, and writing down the vertex weights along this extended path.
Build a RMQ oracle on each of the obtained arrays and, for each $v$, keep a reference to the position of $v$ in the (unique) array associated with the path $P'_v$ such that $v$ is contained in $P_v$.
The total size of the arrays is $O(n)$ and hence the total size of the RMQ oracles is $O(n \log n)$.
A query on an ancestor-descendant path can now be answered in $O(1)$ time by first looking at $M[v, j]$ and a then querying the oracle associated with $P'_{z'}$, where $z'=z_{2^\ell}$.
This is because $L-2^\ell < 2^\ell$ (by the choice of $\ell$), showing that $P_{z'}$ has length at least $2^\ell-1$, and hence all the remaining $L-2^\ell$ proper ancestors of $z'$ (that were not already considered in $M[v, j]$) must be in $P'_{z'}$.
There is a great paper explaining this technique that combines the lookup table $M$ with this extended path-decomposition.
With some additional work you can even get rid of the $O(\log n)$ factor in the space complexity.
