Short answers, pulled from the story.
John Lennon wrote Lucy in the Sky with Diamonds after his three-year-old son Julian showed him a nursery school drawing he called "Lucy - in the sky with diamonds", depicting his classmate Lucy Vodden. Lennon said he thought it was beautiful and immediately wrote the song about it. He also drew on Lewis Carroll's Alice in Wonderland books for the song's fantastical imagery.
Lennon repeatedly denied that the title was an intentional reference to LSD. He said it was "purely unconscious" and that he never noticed the initials spelled LSD until someone pointed it out. McCartney supported this account, though he confirmed in a 2004 Uncut magazine interview that drugs did influence some Beatles compositions at the time.
The BBC never officially banned Lucy in the Sky with Diamonds. The Oxford Handbook of Music Censorship consulted the BBC's own internal correspondence from 1967 and found no ban on the song. The song was broadcast on BBC Radio as early as the 20th of May 1967 on Kenny Everett and Chris Denning's programme Where It's At.
The Beatles began recording Lucy in the Sky with Diamonds with rehearsals in Studio 2 at Abbey Road on the 28th of February 1967. The instrumental backing was finished the following evening, and Lennon's double-tracked vocals were recorded on the 2nd of March. The final mono mix was completed on the 3rd of March 1967.
Elton John released a cover of Lucy in the Sky with Diamonds on the 15th of November 1974, recorded at Caribou Ranch. John Lennon appeared on the recording under the pseudonym Dr. Winston O'Boogie, contributing backing vocals and guitar. The single reached number one in the United States for two weeks in January 1975 and held number one in Canada for four weeks.
Lucy Vodden was the nursery school classmate of Julian Lennon whose name inspired the song's title. She lived in Surbiton, Surrey, and died on the 28th of September 2009 from complications of lupus at the age of 46. Julian Lennon had been informed of her illness and renewed their friendship before her death.