  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.
