"The Covenant of the League of Nations, in spite of the flattering choice of Geneva as the headquarters of the organization, was not regarded with satisfaction in Switzerland. The guarantees for the maintenance of peace in Europe were less adequate than the Swiss project had suggested, and it was obviously difficult for a permanently neutral state to be associated with the enforcement of collective security and with active resistance to aggression. But Swiss idealism welcomed the general implications of the League, and outstanding difficulties were solved when, by the Declaralion of London (Feb. 13, 1920), the council of the League recognized the special position of Switzerland as permanently neutral and agreed that no military assistance, direct or indirect, could be required from it in the event of a breach of the covenant, although 'economic sanctions' would become applicable. The issue was long debated; the federal council advised entry into the League, which was accepted by the Nationalrat [national council--DT] by 115 votes to 50, by the Standerat [council of states--DT] by 30 to 6. The necessary referendum on May 16, 1920 revealed even greater caution; 11 1/2 states were for, 10 1/2 against.." Encyclopedia Britannica, 1957 edition, article "Switzerland."
See
https://en.wikipedia.org/wiki/Swiss_referendums,_1920 for the results of the referendum. It got a clear popular majority (56.3-43.7) but in addition it needed a majority of cantons, and as indicated above it just barely got that (11 1/2 to 10 1/2).
Suppose the referendum had fallen short of getting a majority of cantons. It's hard for me to see the League having its headquarters in Geneva if Switzerland doesn't join the League. Do the headquarters remain in London, or is some other place found? (In OTL, "On 1 November 1920, the headquarters of the League was moved from London to Geneva, where the first General Assembly was held on 15 November 1920."
https://en.wikipedia.org/wiki/League_of_Nations) And are there any other effects of Swiss non-membership?
There is btw an interesting discussion of the Swiss debate on the League at
http://encyclopedia.1914-1918-online.net/article/politics_and_neutrality_switzerland It is even more misleading in Switzerland than in the US to see the debate as being between isolationism and internationalism:
"The vehement opposition of the generally internationalist social-democrats was striking. With the exception of the religious socialist Leonhard Ragaz, they saw the League of Nations simply as an imperialistic order created by the victor; they revised their stance only in the face of rising fascism. On the opposite side, most of the farmers – under the leadership of their long-time secretary, Ernst Laur (1871-1964), who perceived the League of Nations as a safety mechanism against Bolshevism – supported joining it.
The visionary force of many remarks made by the League's supporters, even from unlikely supporters, is impressive. Thus, the yearning for peace by the farmers' leader, Ernst Laur, matched that of antimilitaristic Ragaz, with whom he otherwise had very little in common.[40] The social-democratic journal Volksrecht wrote sardonically against such visions on 15 January 1920 that 'the frenzy of phrases about world peace, uniting all peoples and securing their happiness' recall the times of the Holy Alliance, that 'world federation of reactionaries'.[41]...
It is hardly surprising that the suggestion to join the League of Nations was strongly objected to in German-speaking parts of Switzerland. With the exception of Lucerne, Obwalden, Nidwalden, Appenzell Ausserrhoden and Thurgau, all the German-speaking cantons opposed joining the League. The French and Italian-speaking cantons, in contrast, were overwhelmingly for it; the mixed ones (Freiburg, Valais) were just as clearly in favour of joining; the cantons with language minorities (Bern, Grison) also supported joining, with a smaller majority in favour of entering the League. The majority was clear, by almost one-hundred thousand votes. However, the accession to the League (and implicitly to a differential neutrality) was approved by the smallest possible majority of cantons; this fact led to low expectations for the future, which became tumultuous faster than anyone would have expected, considering the overwhelming desire for peace at the war’s end."