We provide a conformal field theory (CFT) description of the probabilistic model of boundary effects in the wired uniform spanning tree (UST) and its algebraic content, concerning the entire first row of the Kac table with central charge $c=-2$. Namely, we prove that all boundary-to-boundary connection probabilities for (potentially fused) branches in the wired UST converge in the scaling limit to explicit CFT quantities, expressed in terms of determinants, which can also be viewed as conformal blocks of degenerate primary fields in a boundary CFT with central charge $c=-2$. Moreover, we verify that the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs (i.e., Virasoro degeneracies) of arbitrary orders hold, and we also reveal an underlying valenced Temperley-Lieb algebra action on the space of boundary correlation functions of primary fields in this model. To prove these results, we combine probabilistic techniques with representation theory.