A new composite with a core–shell structure based on magnesium hydride and finely dispersed aluminum powder with an aluminum oxide shell was mechanically synthesized. We used magnesium chips to produce magnesium hydride and aluminum wire after exploitation to produce nano-sized aluminum powder. The beginning of the hydrogen release from the composite occurred at the temperature of 117 °C. The maximum desorption temperature from the MgH2-EEWAl composite (10 wt.%) was 336 °C, compared to pure magnesium hydride—417 °C. The mass content of hydrogen in the composite was 5.5 wt.%. The positive effect of the aluminum powder produced by the electric explosion of wires method on reducing the activation energy of desorption was demonstrated. The composite’s desorption activation energy was found to be 109 ± 1 kJ/mol, while pure magnesium hydride had an activation energy of 161 ± 2 kJ/mol. The results obtained make it possible to expand the possibility of using magnesium and aluminum waste for hydrogen energy.