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.