An upper-class Russian child's reading at VN's times (1900s) had to
include Perrault's (for which VN read French originals) as well as
Grimm's, Hoffmann's and Andersen's stories in wonderful Russian
translations. My Soviet generation (1950s), which had only Russian
[very rarely English], still read all of these classical fairytale
authors in various translations.
"Nuts" that are cracked by the Nutckracker in Russia (i.e. in Petipas
ballet libretto set in St. Petersburg) could be both hazelnuts and
walnuts.
Hoffmann's original tale does not specify it, but since Fritz breaks
Nutcracker's jaw (three teeth at once!) with a nut that is way too big,
those are probably walnuts.
Hazelnuts (filbert, Corylus avellana, in Russian called forest nuts,
oreshnik, or leshchina) are native to most of European Russia. Walnuts
(English walnut, Juglans regia; in Russian called Greek nuts), are
native (or introduced long ago) in southern European Russia, Ukraine,
Crimea and Caucasus.
These two types of nuts (orekhi) were (and still are) very common in
Russia but most (if not all) Russian folklore references are to
hazelnut.
Hazelnut was a sacred tree for southern Slavic tribes, a connection to
chthonic world, a talisman protecting from evil; hazelnut was used in
reproductive magic (placed in dowry chests), in erotic riddles ("I have
a furry thing, which has a smooth thing inside, which has something
sweet inside") etc. (http://www.pagan.ru/o/oreshnik0.php)
These magic allusions obviously come from shell/kernel dichotomy, and I
am sure there are volumes written about this just like about egg magic.
By the way, In Russian, the word 'orehi' (nuts) does not serve , as it
does in English, as a substitute for 'testicles'; in this case the
equivalent is 'eggs' [This makes a great joke of Soviet times fully
translatable -- What is a Leonid [Brezhnev] pastry? It is just like
Napoleon [a famed sweet layered pastry], only without nuts ['eggs' in
Russian].
Speaking about hazelnuts and squirrels, the most famous connection of
course is in Pushkin's "The Tale of Tsar Saltan" (also mandatory
childhood reading) with a guarded magic squirrel on a magic island that
cracks precious nuts -- probably hazelnuts since (a) common Russian
squirrels do not live next to walnuts' and (b) walnut does not have a
single 'kernel'.
"Here's a wonder, though, worth telling -
There's a little squirrel dwelling
In a fir tree; all day long,
Cracking nuts, it sings a song.
Nuts-most wondrous to behold!
Every shell is solid gold;
Kernels - each an emerald pure!
That's a wonder, to be sure."
(transl. Louis Zellikoff)
I apologize if these connections are already well-known to
Shadeologists and Pninographers; at least I could not find any
reference to those items in Nabokov-L archives.