Let G be either the Grigorchuk 2-group or one of the Gupta-Sidki pgroups. We give new upper bounds for the diameters of the quotients of G by its level stabilisers, as well as other natural sequences of finite-index normal subgroups. Our bounds are independent of the generating set, and are polylogarithmic functions of the group order, with explicit degree. Our proofs utilize a version of the profinite Solovay-Kitaev procedure, the branch structure of G, and in certain cases, existing computations of the lower central series of G.