We show that hyperbolic electromagnetic metamaterials, implemented as multilayers based on two material constituents arranged according to the Thue-Morse aperiodic sequence, may exhibit strong nonlocal effects, manifested as the appearance of additional extraordinary waves which are not predicted by standard effective-medium-theory (local) models. From the mathematical viewpoint, these effects can be associated with stationary points of the transfer-matrix trace and can be effectively parametrized via the trace-map formalism. We show that their onset is accompanied by a strong wave-vector dependence in the effective constitutive parameters. Despite the inherent periodicity enforced by the unavoidable (Bloch-type) supercell terminations, we show that such strong nonlocality is retained at any arbitrarily high-order iteration, i.e., approaching the actual aperiodic regime. Moreover, for certain parameter configurations, at a given wavelength and for two given material layers, these effects may be significantly less prominent when the same layers are arranged in a standard periodic fashion. Our findings indicate that the (aperiodic) positional order of the layers constitutes an effective and technologically inexpensive additional degree of freedom in the engineering of optical nonlocality.