Once upon a time, there lived a young maiden who grew up amid the soot and grime of central London. She was a mischievous sort, playing on World War II bomb sites when she wasn’t furtively listening to her father’s record collection, scratching up his Handel and Beethoven albums in the process. She was often punished for such disobedience.

“The worst time was when I was sent away to a private girl’s school in the deepest countryside at the age of 14,” the British folk singer Vashti Bunyan says from her home in Edinburgh via email. “I was horribly homesick for London and for freedom.”

If Ms. Bunyan’s story begins like a fairy tale, the difficulty may be in discerning whether her story hews closer to that of Cinderella or Sleeping Beauty. The former fits in that this 61-year-old folk singer will be making her Carnegie Hall debut Friday as part of the David Byrne-curated “Perspectives” series, subtitled “Dreamland,” 43 years after she started singing in London nightclubs. And yet the latter heroine may be more apt to what transpired in the intervening years.

“Being a teenager in the early ’60s was like pioneering, all uncharted, exciting,” she said. “Now the ’60s are seen mostly for their psychedelic years. But what took place before that — brave rebellion in the face of severe authoritarianism at the start of the decade — that was what paved the way.”

In 1964, Ms. Bunyan was a student at Oxford’s prestigious Ruskin School of Drawing & Fine Art (alongside future Monty Python alums Terry Jones and Michael Palin) when an early Rolling Stones show forced her to reconsider her path. “[I] fell for Mick Jagger immediately. He was so scornful and aloof, and the music was so raw and different and wild.”

It wasn’t long before the school expelled her for concentrating on music instead of art, so she picked up a guitar and headed back to London. “I sang in a few dingy nightclubs … there in my scruffy black clothes, just me and a guitar and some sad little love songs,” she said, quick to clarify that “I never sang in folk clubs; I was not a folksinger.”

She auditioned for Stones manager Andrew Loog Oldham, who was seeking a female singer to replace the recently signed Marianne Faithfull on his roster. Soon after, Ms. Bunyan cut a Jagger-Richards number called “Some Things Just Stick in Your Mind,” her quiet voice swaddled in orchestral bombast.

Chart success failed to materialize, and by 1968 Ms. Bunyan and her boyfriend, Robert Lewis, had reached the breaking point. “I remember feeling desolate … and the world outside was making less and less sense … but I could not really identify my feelings of hopelessness,” she said. “I wanted to somehow get to the realities of life and try to live differently than the way I had been brought up. I also wanted to get as far away from London as I could, for it was filled with musical failure for me.”

Hitching a cart to a horse, she and her boyfriend set off for an artist commune spearhead by the folk singer Donovan on the Isle of Skye. It was an arduous trek through the elements, but it opened Ms. Bunyan up to the natural world. “I started writing songs about what I saw and telling the story of our road life. They were just a kind of diary. I was quite sure no one would ever hear them except Robert and myself. And the horse, Bess.”

The producer Joe Boyd, who gained renown capturing seminal British folk acts like Fairport Convention and the Incredible String Band, as well as folkie Nick Drake, persuaded Ms. Bunyan to return to London in the winter of 1970 to record “Just Another Diamond Day.” Aided by members of the aforementioned bands, as well as Drake’s arranger, Robert Kirby, Mrs. Bunyan was nonplussed by the proceedings. “[Joe] wouldn’t let me use any overdubs or trickery of any kind, which annoyed me a bit,” she said. “I wasn’t the same kind of purist as he was. I wanted the orchestras and the big arrangements.”

“Diamond Day” is instead a bucolic collection of folk songs delivered in a voice as warm as a campfire and as hushed as dawn’s first murmurs. But at the time, Ms. Bunyan fretted about its unadorned folkiness. “I was worried about the handmade sound of it, all the wrong notes that I would have taken out that Joe left in.”

Such second-guessing was only reinforced by the album’s quick disappearance from the marketplace, and Ms. Bunyan hung up her Martin guitar resolutely. Returning to her Bess-drawn cart, she decamped throughout Ireland and Scotland while raising her three children for the better part of three decades.

Around 2000, when Drake’s “Pink Moon” scored a Volkswagen commercial and brought about his renaissance, Ms. Bunyan’s lone album was reissued, and its heartfelt honesty and unadorned vision caught the ears of a new generation of listeners. It wasn’t until she stumbled upon mentions of her name on the Internet that she realized “Diamond Day,” with all its wrong notes and handmade sounds, had taken hold in the 21st century.

A new wave of ardent admirers encouraged Ms. Bunyan, and she stirred like Sleeping Beauty. After about 35 years of silence, she began appearing on recordings by fans like Devendra Banhart (also on the Dreamland bill) and Animal Collective before releasing her critically acclaimed second album, “Lookaftering,” in 2005.

Although success eluded her the first time around, Ms. Bunyan is back singing before enraptured audiences, appreciating every minute of the fairy tale. “It’s good to be back,” she said, “more than I can ever say.”