Hydrogen absorption in magnetic thin film nanostructures can modulate their electronic, magnetic, and transport properties by modifying the electronic structure and lattice strain. However, the influence of composition and nanostructuring on these two competing effects is not well understood. We systematically investigate hydrogen-induced magnetic hysteresis in Co$_x$Pd$_{100-x}$ alloys, [Co(0.1 nm)/Pd(d)]$_{15}$, and [Co(0.2 nm)/Pd(d)]$_{15}$ multilayers, using extraordinary Hall effect characterizations (EHE) in air and 4% H$_2$/N$_2$ mixture. We show that the hydrogen-induced response is not universal, but depends strongly on composition and layer thickness. This reflects competition between Pd-related electronic effects and magnetoelastic anisotropy. In Pd-rich CoPd alloys and Co(0.2 nm)/Pd multilayers, hydrogen initially contracts the hysteresis loops at low Co fractions, followed by loop expansion above x ~ 40%. This contrast results from competition between suppression of Pd-induced magnetization through Pd-4d band filling and hydrogen-driven anisotropic strain that strengthens magnetoelastic anisotropy in Co-rich samples. In contrast, in ultrathin [Co(0.1 nm)/Pd(d)]$_{15}$ multilayers, hydrogen induces a weak, non-monotonic but generally expanding loop behaviour across x = 15-60%, indicating a dominant role of interfacial magnetic connectivity and strain-mediated magnetoelastic anisotropy in the ultrathin limit. Furthermore, we observe a hydrogen-induced reversal of the EHE loop polarity near the crossover regime, reflecting a change in the dominant EHE scattering mechanisms, thus providing an additional degree of magnetic tunability by hydrogen. These results demonstrate that hydrogen can selectively tune the magnetism of CoPd nanostructures via composition-controlled electronic and magnetoelastic effects, offering insights for hydrogen-responsive spintronic and sensing devices.