The Beatles never did learn to read or write music using traditional notation, and by all accounts were happy with that fact. Music was a discovery process for them that did not involve any books -- they once traveled across town for someone to teach them a B7 guitar chord. John Lennon's mother taught him banjo chords; they lasted with him through his early Quarry Men days until Paul McCartney showed him proper guitar chords. George Harrison learned guitar through lessons and painstaking practice. To acquire new material for concerts (mostly cover songs in the early days), they learned largely through listening to records and mimicking the sounds as closely as they could.
When it came to official transcriptions of their music, the Beatles could provide input if necessary, but the transcribing was left to others (e.g. Question: Which of the vocal melodies in "Baby's in Black" is the lead? Answer from Paul: Both). When classical/professional musicians were employed during recording sessions, the Beatles would often sing the melodies and George Martin would transcribe them into sheet music.
For their part, the Beatles liked not knowing how to read sheet music. George Harrison once remarked that he thought if he learned too much of music theory it would ruin the songwriting process in terms of having an innovative ability. Would John have written the strange time signatures of "Good Morning Good Morning" or Paul the interesting modalities of "For No One" as instinctively having the full knowledge of music theory at their fingertips? Perhaps not.