I’ve looked into the code for the effects of the following cards, and none of them affect Hospital or Commandery. Maybe the developers overlooked it or it’s intentional for some reason.
TEAM Art of War: (Chinese) War Academies, Barracks, Stables, Corrals, and Caravanserais cost -30%; Blockhouses, Nobles’ Huts, and War Huts cost -10%
TEAM House of Romanov: (Russians) Barracks, War Camps and War Academies get +100% hit points; Blockhouses, Kallankas, War Huts, and Nobles’ Huts get +50% hit points; Watch Towers get +10% hit points
TEAM Reichsritter: (Germans) Stables, Corrals, and Caravanserais cost -50%; Nobles’ Huts, War Academies, and War Camps cost -25%; Kallankas cost -15%
TEAM Spanish Road: (Spanish) Barracks cost -50%; War Academies and War Camps cost -25%; Blockhouses, War Huts, Nobles’ Huts and Kallankas cost -15%