Recent studies on fully dielectric multilayered metamaterials have shown that the negligibly small nonlocal effects (spatial dispersion) typically observed in the limit of deeply subwavelength layers may be significantly enhanced by peculiar boundary effects occurring in certain critical parameter regimes. These phenomena, observed so far in periodic and randomly disordered geometries, are manifested as strong differences between the exact optical response of finite-size metamaterial samples and the prediction from conventional effective-theory-medium models based on mixing formulae. Here, with specific focus on the Thue-Morse geometry, we make a first step toward extending the studies above to the middle-ground of aperiodically ordered multilayers, lying in between perfect periodicity and disorder. We show that, also for these geometries, there exist critical parameter ranges that favor the buildup of boundary effects leading to strong enhancement of the (otherwise negligibly weak) nonlocality. However, the underlying mechanisms are fundamentally different from those observed in the periodic case, and exhibit typical footprints (e.g., fractal gaps, quasi-localized states) that are distinctive of aperiodic order. The outcomes of our study indicate that aperiodic order plays a key role in the buildup of the aforementioned boundary effects, and may also find potential applications to optical sensors, absorbersand lasers.