I think this is a huge stretch. Sorry. WW1 was not caused by Nietzsche's philosophy. In fact I would be willing to bet that all of the major protagonists were self-avowed Christians and few had even read the mad Pole, who, after all was not nearly as much a household name then as now. The author really offers no details on the connection between Humanism and everything that followed. To make the strong cause and effect case he points out I think he'd need to do a little more.
Nietzsche was a German and he was very well known by the German public befrore and during WW1. Many German soldiers carried small copies of Nietzche with them.
That would likely be true with respect to Catholic dominated Austria and Hungary, Lutheran dominated Prussia/Germany, Anglican dominated Britain, and Orthodox Russia and Serbia. However, both France and Italy had undergone anticlerical periods prior to the outbreak of World War I. The Catholic Church, which lost the Papal States as part of the unification of Italy, did not recognize the legitimacy of the Italian government. In part due to reaction against the persecution of Alfred Dreyfus, France had entirely severed church-state ties. Although in 1914 the majority of Frenchmen and Italians were Catholic, the leadership of those nations was anticlerical.