This story bears characteristic signs of a tall tale, although in this case one can identify the origin. It appears to be an amalgamation of two anecdotes, neither of which is itself very credible. Both are traceable to Warren McCulloch, Pitts's co-author on "A Logical Calculus of Ideas Immanent in Nervous Activity" (1943), which proposed the first mathematical model of a neural network. There is some vague connection to actual events, which might have prompted the story's retelling.
"He is widely remembered to have spent three days" reading Principia in the library, says Wikipedia, without naming any sources of this wide rememberance. Upon closer examination the story is interpolated from Talking Nets by Anderson-Rosenfeld, not exactly a historical work, and the more reputable Smalheiser's short biography of Pitts. Both name McCulloch as the anecdote's source, Smalheiser even references his Collected Works:
"The story is told that, at age 12, Pitts ran into the public library to hide from some bullies, found a copy of Principia Mathematica by the 20th-century philosophers Bertrand Russell and Alfred North Whitehead, and proceeded to read it cover to cover in the next few weeks . Pitts experienced a metaphysical insight that logic rules the universe, and as a corollary he felt that ego — and his ego in particular — needed to be erased in order to achieve an understanding of the world."
There is no mention of the letter here, and "the story is told" is an odd way to introduce a story one believes. He then quotes the other story, about Pitts meeting "Bert" in a park in Chicago when he ran away from home at 15 (Anderson-Rosenfeld make him 14 for that). And then
"Now, this Bert talked with Walter for some time of philosophy and mathematics, and came to realize that this was no ordinary youngster. Bert was impressed. He told the boy that Carnap, then Professor of Philosophy at the University of Chicago, had written a book that would interest
him, and urged him to go and speak to the grand old man. So, Walter got himself
a copy of Carnap’s book and read it. Later, Carnap was to recount the meeting thus: “This young boy came in to see me and said he had read my book and that a certain paragraph on a certain page was not clear to him..."
Guess who told this story. It was Blum with reference to... McCulloch. This time Smalheiser is even more explicit about its credibility:
"Though the story of meeting “Bert” in this manner may be apocryphal, it
is true that Pitts sat in on Bertrand Russell’s course when the latter visited
Chicago in 1938, and that he walked into Rudolf Carnap’s office with a marked copy of his book filled with corrections and suggested improvements. After his initial contact with Carnap, he disappeared, and almost a year went by before Carnap was able to contact him again, supervise some
of his studies, and help him secure a student job."
In 1938 Pitts was indeed 15, but there is no mention that the marked copy of Carnap's book survived, however. And even if we are to believe McCulloch's recollections, the alleged letter, let alone Russell's reply to it, are additions by later "contributors".