Prime Localizations of Burger--Mozes-type groups

This article concerns Burger--Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primes $p$ we determine their local $p$-Sylow subgroups as well as subgroups of their $p$-localization, which is identified as a group of the same type in certain cases.