Anouk Aimée had her stage name from her character in La maison sous la mer, by Henri Calef (1947), when she was thirteen years old. The film was based in an homonyme book by Paul Vialar published in 1941, so the name was known in France since this year (the popularity of Anouk is due to Anouk Aimée's success, of course).
Vialar's story is based in Normandy and it is very possible that the name was used there as nickname of Anne, because other endings in -ic/-ick/-ig, -ou/-on/-oun (spellings vary depending the source)... are very usual in nicknames in Normandy and Brittany. I don't know if the existence of the Russian nickname Anouchka could have influence in the creation of the nickname (perhaps via Anouchka>*Anouch>Anouk), but the junction of this influence is possible because in France the Russian trend was popular in early 20th century.
At this moment, I don't have evidences of use of Anouk as nickname in Normandy and surronding areas but this is not odd because the nicknames rarely appears in official documents and records and because the French naming law was very restrictive before 1966.