The new film by the Coen brothers, Inside Llewyn Davis, evokes Greenwich Village at the beginning of the American folk boom. The date is February 1961. Metropolitan young Americans sit in smoky clubs listening reverently to music that they believe is purer, more honest and more heartfelt and therefore more elevating than the commercial mainstream of Sinatra, Buddy Holly and Doris Day. Folk music is still mainly a process of discovery and renewal rather than invention; singers tend to see themselves as curators of tradition. Lines such as: "Here's a song I first heard Leadbelly sing," remain the staple fare of introductions in a form that awaits the great singer-songwriter. Bob Dylan has just arrived in town but is still a nobody to everyone but himself (he comes late in the film, and then only as a portent). Other about-to-be-famous performers – Peter, Paul and Mary, the Clancy Brothers – are easily detectable inside their light fictional coating.

Despite a despairing story and an unsympathetic and sometimes frankly dislikable protagonist, Inside Llewyn Davis is a pleasure to watch. The cinematography is never dull, the script is witty and the characters marvellously well acted, so that their peculiarities are entertaining and persuasive. In at least one historical aspect, however, the film is fundamentally unbelievable. This may not seem to matter – the Coens didn't set out to make a documentary – but please bear with my pedantry, which concerns a beautiful song that became important to the British folk movement, and which in 1961 sounded rather old when it was in fact very new. Early in the film we glimpse its title when we see a 78rpm record that the eponymous Davis (played and sung by Oscar Isaac) had made as a child. His sister tells him he must have been about eight years old. The label says "Shoals of Herring". Later, in what would be the film's tenderest moment, if the Coens ever allowed tender moments to survive, Davis visits his former-seaman father in a home for the old and demented and sings the song, reminding his wordless father that it was once an old favourite. "Oh, it was a fine and a pleasant day/ Out of Yarmouth harbour I was faring/ As a cabin boy on a sailing lugger/ For to go and hunt the shoals of herring."

In February 1961 the song was barely six months old, having been created by the British singer and writer Ewan MacColl for a radio documentary, Singing the Fishing, which was first broadcast by the BBC on 16 August 1960 as the third in a series of eight "radio ballads" celebrating British working-class life. Researching the programme, MacColl met an old Norfolk fisherman, Sam Larner, who'd first gone to sea in 1892. MacColl shaped his interview into lyrics and set them to an old folk tune, Shores of Erin; the result was a MacColl composition that over the next decades took its place with The First Time Ever I Saw Your Face and Dirty Old Town as one of his most celebrated songs. An LP that included the song and some of Larner's interview appeared in 1961, and it seems impossible that anyone living in America could have heard the song before then. As an old bond between father and son, Love Me Do would be only slightly more unlikely.

The "unauthentic" is the traditional enemy of folk music, and it may be (though I don't believe it) that the Coens have given The Shoals of Herring a place in their film as a sly joke about the business of knowing what's true and what isn't. How "authentic" can a song about herring fishing be if it was never actually sung by fishermen? What happens to authenticity when, in pursuit of some idea of it, Robert Zimmerman of Hibbing, Minnesota, becomes Bob Dylan of Greenwich Village – "inventing a character to go with his songs" in the words of Joni Mitchell, though Dylan himself saw it as a simpler matter of some of us getting born "to the wrong names and the wrong people". Nor was he alone. Jimmie Miller, born to Scottish parents in Salford, Lancashire, became Ewan MacColl, born in Auchterarder, Perthshire, when he was over the age of thirty. "I never knew what all that Scottish business was about," his wife of the time, Joan Littlewood, once told me, but it obviously derived from how he wanted to feel about himself and be seen by his audience. A paradox: to become more authentic, MacColl became less so. He went further than Dylan in inventing his birthplace, but unlike Dylan he worshipped his parents, who were also singers, and kept them in his story.

Authenticity was the rod that folk music made for its own back. Industrialism and the rise of nationalism in 19th-century Europe fuelled the search for music that could be called national and traditional, easily made and enjoyed by ordinary people, with a pastoral history that predated factories, music halls and professional musicians. In England, the musician and composer Cecil Sharp collected the songs and dances of the rural working class, but what became known as the first British folk revival was soon criticised for its quaint gentility (for example, how typical – or authentic or historic – were the music and dancing Sharp recorded at an Oxford quarry in 1899, which started the cult of morris dancing?). Salvation came in 1948 when the first copies of The People's Song Book reached Britain from America to give British folk music its modern identity. The songs included We Shall Not Be Moved, Joe Hill and Strange Fruit – songs that had been tested "in the fire of the people's struggle all around the world" wrote the anthology's editor, Alan Lomax, offering them as "a folio of freedom folklore, a weapon against reaction, and a singing testament to the future."

Lomax's words inspired MacColl, who was an enthusiastic communist, and he and his associate Bert Lloyd reinvigorated folk music with songs about strikes and suffering that contrasted strongly with the old folklorist view of Merry England or Bonny Scotland. In the view of one of MacColl's communist mentors, all music ultimately originated in "the rhythmical movement of human bodies, engaged in collective labour". A lot of popular music tended to deny this idea; folk music therefore existed to combat capitalism's "brain-softening commercial culture", which the music of the world's greatest capitalist nation especially exemplified. Under the leadership of MacColl and Lloyd, the British folk movement became purist and nationalist in its fight against American imperialism, apparently blind to the fact that it owed its original energy to American music. Instructions went out to folk clubs that singers should stick to songs of "their own native tradition". Successful popular performers such as Lonnie Donegan and, later, Dylan, who had come out of folk and skiffle traditions were detested as turncoats. The cry of "Judas!" that surprised and amused Dylan at his Manchester concert in 1966 came out of this bewildering struggle to cling to the allegedly authentic.

Folk music flourished despite it all, and in Britain rather more than America. In the early 1960s, we could watch Robin Hall and Jimmie MacGregor most weeknights on the BBC, while series with gee-ha names such as Barndance, Hullabaloo and Hoot'nany would bring us singers who held their hands to their ears. In this broader world, nobody bothered much with manifestos or the need to avoid compromise. When in 1962 I found myself singing "Freight train, freight train, goin' so fast" with a couple of schoolfriends to an audience of Fife miners – real miners, before they lived only in songs – the complicated history of the song (it was written by a black housemaid of the Seeger family) was known by none of us. Folk was becoming mainstream. Still, our audience that night – our only audience ever – didn't like it. "Here's a song I first heard Nancy Whiskey sing." I wish I could have said that.