Nabokov was born in 1899. Wikipedia says, "The family spoke Russian, English, and French in their household, and Nabokov was trilingual from an early age. In fact, much to his patriotic father's chagrin, Nabokov could read and write in English before he could in Russian." It also mentions that, despite living in Berlin from 1922 to 1937, he never really knew much German.
My understanding was that English's ascent in the international scene was a product of World War I, with Britain and America ascendent, and was solidified in the post-WW2 era when very mono-lingual American were footing a lot of the bill for much of the reconstruction of Europe and then much of the defense of market economies in the rest of the world.
All the important physics breakthroughs of this period were published in German (though done by primarily by German and Danish scientists). I know that in the 19th century, that a German and a Russian diplomat might well correspond in French, and I assume that would continue for into the first part of the 20th century. But what was the role of English in the international scene before 1918? I know people were learning it--James Joyce taught English in Trieste from 1904 to 1920--but what associations did it have during this period? Because of Britain's large empire and the Industrial Revolution was it associated with trade and perhaps engineering? Was English seen as a primarily a practical language? Or did it have romantic political associations, because of the long Anglo-American heritage of democracy and Anglo-Scottish-American liberal thought (Locke, Hume, Adam Smith, John Stuart Mill, John Dewey, etc)?
Regarding the lingua franca of science at the time, Ammon 2012 reports that in 1910 French, English, and German were used approximately the same for scientific publishing, with German at 32% of the total, English at 30% of the total, and French at 25% of the total. You'd really have to be looking at particular disciplines here, because some had one language being used in them more than the others, for example German in chemistry and philology/linguistics.
If anyone wants a nice read on the subject, they should check out Gordin 2015 - Scientific Babel.
According to his cousin Ivan Nabokov Nabokov had an English nanny. This was not at all unusual in wealthy households across 19th-century Europe, and became especially prestigious as Queen Victoria was personally involved in selecting British nannies for her various royal grandchildren and great-grandchildren. It was not that English was considered an important language to learn, but rather that Englishwomen (and Scotswomen and Irishwomen) were considered the best at caring for infants.
Charlotte Zeepvat's From Cradle to Crown goes into greater depth on this.
You will find that Lermontov and Pushkin were extremely fond of Byron and some of the people featured in War and Peace had English nicknames. Pushkin also translated the Pilgrim's Progress; Keats and T.S.Elliot are cited in the 20th century by Anna Akhmatova and she devotes some of her poetry to Isaiah Berlin. The Anglos had presence in the Russian literary scene and according to Nabokov himself his father was an anglophile and his nanny an Anglo.
However: Take all this with a nice pinch of salt. Russians often note he'd enjoy shooting bull to his American public (f.e. of his ancestor having borrowed books to the imprisoned Dostoyesky in the "lectures on Russian literature") and on this particular occasion it appears to have been a necessity with critics having nothing better to say than that he's not a "native speaker" and thus somehow "awkward" or "unnatural" when writing English. Being American they couldn't imagine anyone speaking more than one language with fluency. Well, what else do you say to people like that?
edit: And the Anglo literati will recognize where Chekhov got his Three Sisters play from ;)