I believe Dr. Evans is correct about the appearance of several of these names in Russian novels which have been widely read in the Western world.
He mentioned
Sonya; another example is
Natasha, who appears as a major character in "War and
Peace." That name was probably boosted by a film version in the 1950s.
Nadia, however, first appeared on the SSA list in 1976 because of the fame of the Romanian gymnast
Nadia Comaneci in the Montreal Olympics.
I am curious about
Tanya, because the most famous Tanyas did not appear until the early 1970s and yet the name was already becoming popular in the 1960s.
Sasha is also interesting because it is a male diminutive in Russian. I suppose it looks enough like a "typical" girl's name in English that parents thought it would be better for girls than boys.
I don't believe any of the other forms mentioned (
Anya,
Anoushka,
Vanya,
Katya) have had noticeable success in the US although they may be more popular in some European countries.