Here's a link to the lyrics Of "Tschaikowsky (and Other Russians":
Digital Inflection: Tschaikowsky And Other Russians (about halfway down the page)
Like other researchers who presumably have taken a shot at your
question, I initially (surprisingly) found no online source for the
lyrics, after using variant spellings of Tchaikovsky's name in my
searches, along with a wide variety of other search terms related to
Danny Kaye, Kurt Weill, Lady in the Dark, and so on.
Knowing that the lyrics were by Ira Gershwin and that there is a
coffee-table book of his complete lyrics, I then checked Amazon.com to
see if that site's "Search Inside the Book" feature is available for
that book. It is, and I found the lyrics by using it. Here is its
listing page for the book, in case you are interested:
Amazon.com: The Complete Lyrics of Ira Gershwin
I whiled some time away attempting to decipher the names of all of the
50 Russians at the Amazon site, since I have Kaye's recording of this
tour de force patter song, which goes by so fast that some of the more
obscure names are gibberish to a non-Russian listener.
It then occurred to me to try one more Google search using a string of
those names instead of focusing on "Tchaikovsky" with all its variant
spellings and well over a million online references.
This resulted in locating the site linked above, which, I am fairly
confident, is the only online source of these lyrics other than the
Amazon.com site, where the scanned page (page 293) from the book is
almost illegible. This is the simple search that worked:
Notice that Ira Gershwin spells the composer's name as "Tschaikowsky,"
which is much less common that the usual English spelling,
I am confident that this is the exact information you are looking for.
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