I would suggest French and English as the languages of 'international' cooperation; although French's "virtual monopoly" as diplomatic language would wane in the sense that interpretation became more common at conferences (e.g the International Labour Organization), it was certainly still one of the two working languages adopted. English was the other, having an equal footing. From the Standing Orders of the first Labour Conference in 1919:
The French and English languages shall be the official languages of the conference. Speeches in French shall be summarized in English and vice versa by an interpreter belonging to the secretariat of the conference. A delegate may speak in his own language, but his delegation must provide for the translation of a summary of his speech into one of the two official languages by an interpreter attached to the delegation. The summary thus translated will then be rendered in the other official language by an interpreter belonging to the secretariat.
As the minutes make clear, the use of other languages was not always favoured, and there was an expectation to stick to the bilingualism. As the Latin American delegates found out the hard way (the annotations here are from Baigorri-Jalón:
The PRESIDENT [Barnes, G.B.]: [After an intervention presumably in Spanish, although the record does not state so] While the interpreters are getting the hang of this [proposals of amendments in the language of a resolution] let me say that I hope if there are any further Spanish speakers who can speak French, they will put their speeches in French and save the time of the conference. […]"
Yikes!
That being said, our concept of "the international" might not just be limited to formal bureaucratic conferences or international organisations. What languages were spoken within transnational networks were affected greatly by the center of gravity bringing them together, and in many cases, imperial/anticolonial/racial circumstances. My impression is that the Pan African Congresses (the 1st of which occurred the same year as the Labour Conference) had its proceedings primarily in English, with translation for many delegates; nonetheless, its Black Review publication was released in French, English, Spanish, and Portuguese, reflecting the background of delegations and where their struggles were concentrated. Michael Goebel's excellent book Anti-Imperial Metropolis traces anti-colonial and liberation figures in Paris in the interwar period, ranging from Ho Chi Minh to Lamine Senghor, from a small pool of Chinese migrants to Algerian bricklayers: chapters 7 ('a revolutionary lingua franca') might be of interest to you. A fascinating story of unlikely connections made possible by shared languages, (literal) crossing of paths, cafes, jazz and student gatherings.
I'd disagree with /u/Radman1804 that French fell out as the common language by the 40s: at the founding of the United Nations, it seemed quite commonsense to diplomats that French and English should be the working language. I wrote a bit about this period in response to a question by /u/hellcatfighter sometime ago.
References:
Jesús Baigorri-Jalón, 'Conference Interpreting in the First International Labor Conference'
Michael Goebel, Anti-Imperial Metropolis: Interwar Paris and the Seeds of Third World Nationalism (New York: Cambridge University Press, 2015).
Clarence Contee, 'Du Bois, the NAACP, and the Pan-African Congress of 1919'