We consider limit theorems for a random walk on a weighted Galton-Watson tree, where the edges of the tree are assigned randomly uniformly elliptic conductances and the random walker crosses an edge with probability proportional to its conductance. For this process, the effective speed or variance depend in an intricate way on the law of the conductances. In order to study this dependence, we assign to a positive fraction of edges a small conductance $\varepsilon$. We show a functional central limit theorem for the distance of the walker to the root and, provided that the tree formed by larger conductances is supercritical, we show that the variance is nonvanishing as $\varepsilon\to 0$, which implies that the slowdown induced by few edges with small conductance is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in the edge weights.
