Now that the texts of all of the Plenary talks at the international congresses of mathematicians is available online, that could serve as a data source for the transition. This year, every plenary talk was in English, and none of the speakers (that I attended) were at all difficult to understand, at least not on account of the language.
Here are some years, and the number of Invited plenary Speakers whose title is in (English, French, German, Russian).
1920: ( 2, 3, 0, 0), in Strasbourg
1932: ( 2, 9, 9, 1), in Zurich
1936: ( 7, 2,12, 0), in Oslo
1950: (19, 2, 1, 0), in Cambridge
1954: (14, 3, 2, 1), in Amsterdam
1958: (13, 3, 2, 1), in Edinburgh
1962: ( 9, 3, 1, 3), in Stockholm
1966: ( 9, 1, 2, 5), in Moscow
1970: (15, 1, 0, 0), in Nice
1974: (15, 2, 0, 0), in Vancouver
1978: (17, 0, 0, 0), in Helsinki
1983: (12, 0, 0, 0), in Warsaw
1986: (13, 0, 1, 1), in Berkeley
1994: (14, 0, 0, 0), in Zurich
1998: (15, 0, 0, 0), in Kyoto
2002: (19, 1, 0, 0), in Beijing
2006: (19, 0, 0, 0), in Madrid
So the supposition that World War II is the cause isn't supported by this data. Certainly it impacted the use of German, but other languages were common until the 1970s. Something drastic seems to have happened between 1966 and 1970.
