Being given a copy of Mary Poppins by PL Travers for my eighth birthday was both a thrill and, as it turned out, one of the greatest disappointments of my young life. A thrill, because for the last five years I had lived and breathed the Disney version, which had come out in a blaze of glory in 1964. At school I won the unofficial prize for the person who had seen the film the most times (I said eight, although it was actually only six: but in the Disney universe, believing something hard enough is the key to making it come true). I did, though, definitely win the competition for who could say "supercalifragilisticexpialidocious" backwards. And, at home, I sternly presented my grandmother with the sheet music of the Sherman Brothers' score. Like a bad fairy princess bullying her court musician, I made Grandma play on and on until the moment when it seemed that we might both spin ourselves into a cloud of coloured chalk dust of the kind that Bert the pavement artist uses to sketch his magical alternative worlds.

But the moment I unwrapped my present I knew something had gone horribly wrong. My parents had form when it came to missing the point. Over the years I had been the recipient of endless cover versions of branded toys – knock-off versions of Barbie, Caran d'Ache and Ladybird – and this Mary Poppins looked distinctly counterfeit too. For a start the cover was a muddy, slightly sinister pink. This, I now think, was a nod to "strike me pink", one of Mary Poppins's favourite sayings, or perhaps to the blossom in Cherry Tree Lane. Anyway, pink was not then the absolute obsession with little girls that it has since become, and I had been hoping for the luscious, bleeding colours of Disney's Technicolor. But far worse was the picture of Mary Poppins on the cover. Drawn by Mary Shepard, daughter of EH Shepard who illustrated Winnie-the-Pooh, this Mary Poppins looked nothing like the soft and lovely Julie Andrews. In fact, she resembled a stiff peg doll, thin and hard, with a peg nose and two spots of high colour on her wooden cheeks. This was the kind of nanny, magical or not, from which any sensible child would shrink.

I know now that this hard-cornered Mary Poppins was not some careless slip on Shepard's part – the book preceded the film by 30 years and, as far as Travers was concerned, it was Walt Disney who got it wrong when he added spoonfuls of sugar, not to mention some larky cartoon penguins, to her really rather dark text. It is Travers's and Disney's fraught tussles over whose Mary Poppins would finally triumph that is brought to life in Saving Mr Banks, the new film starring Emma Thompson and Tom Hanks.

Anyway, as Travers's text makes clear, Mary Poppins is no beauty. She has squinty eyes and big feet and regularly attracts the comment from the other characters that she is "not much to look at". Nor does the original Mary Poppins sound anything like the carefully modulated Julie Andrews. Travers gives her the accent and vocabulary of a real London nanny: cockney base notes overlaid with a strangled gentility. So she says things like "I'll have you know that my uncle is a sober, honest, hard-working man!" and punctuates her pronouncements with "a superior sniff".

As I read further into Travers's book my suspicion grew that I had been palmed off with inferior goods. There was no Bert here, not really. He makes a brief appearance as a matchman and pavement artist, but there is no trace of the one-man fun factory that is Dick Van Dyke. Mary Poppins doesn't arrive from the sky, at least not in the first book (there are six in total) and, most discombobulating of all, there are not two Banks children but four: Jane, Michael and baby twins John and Barbara. Oh, and Mrs Banks didn't lark around singing about suffragettes and Mr Banks never got to fly a kite.

But I could have forgiven all these dreadful derelictions – how furious Mr Disney must be! – if the book wasn't so oddly dull. Or perhaps that should be dully odd. There were long, waffly passages in which characters who didn't appear in the Disney film went on and on about the stars, and the wind and the moon and the Grand Chain that connects all creatures. Even the adventures that Mary Poppins undertakes with her charges had an indeterminate quality, without a proper resolution. At the end of each outing Jane and Michael go to bed not sure what has just happened or what it all means. And the same was true of eight-year-old me. Disgusted, or perhaps disturbed, I tucked Mary Poppins on to the "too difficult" section of my small library and moved on to Paddington Bear. Rooted in an earthly geography of Peru, Portobello Road and marmalade sandwiches, you knew where you were with Paddington.

A large part of Travers's quarrel with Disney was that she didn't create specifically for children, whereas that was all he did. One of the best scenes in Saving Mr Banks shows Travers, played by Thompson, overwhelmed with horror to find that her LA hotel room has been stuffed with soft cuddly toys from the Disney Corporation's celluloid bestiary, including Winnie-the-Pooh. Rather, Travers saw herself as a mythographer. A follower of Gurdjieff, the Russian mystic who introduced the west to a ragbag of eastern mysticism in the first part of the 20th century, Travers was more interested in excavating the archetypes that underpinned esoteric Christianity than dreaming up nursery pap. She always made the point that it was the grownups, not the children, who needed Mary Poppins most. One of the most important scenes in the book concerns the baby twins John and Barbara Banks who are able to talk to the sunlight and the wind and to animals and who swear that they will never forget this blissful world of oceanic oneness. But on their first birthday they are thrown out of paradise, just as Mary Poppins always said they would be, and are no longer able to communicate with the natural world. It was to help all the adult Johns and Barbaras find their way back to this place of innocence and grace that Mary Poppins first came to earth.

Rereading the Mary Poppins books today, it is not the cod theology that hits you so much as their economic and political underpinning. While the Disney film is set at the apogee of empire – "The year is 1910, it's the age of men" crowed David Tomlinson as Mr Banks – Travers's book is firmly located in the 1930s, Auden's "low, dishonest decade". The first awful shock of the great depression might have been over, but there's a sense that nothing can ever be relied on again. On the first page Travers tells us that No 17 Cherry Tree Lane is the smallest and shabbiest house in the street because Mr Banks has given Mrs Banks the choice between a large family or "a nice, clean, comfortable house". When Banks arrives home from work each evening he often can't spare any coppers to give to the children because "the bank is broken". And then there's the vexing business of the servants' wages which, the increasingly harassed master of the house declares, are ruining him.

No 17 Cherry Tree Lane is clearly in the midst of what was known by the interwar period as "the servant problem". With young working-class people more and more reluctant to "go into service", the middle classes were obliged to run their houses with the help of a dwindling cohort of increasingly bolshie staff who weren't about to put up with any nonsense from their "betters". And that, surely, is the point Travers is making when Mary Poppins turns up from nowhere and refuses to provide references. Hysterically grateful that anyone at all is prepared to work for her, a cowed Mrs Banks immediately agrees to waive the formalities.

Travers was never precise about her socialist affiliations – "it is difficult for me to think or feel politically" – preferring to dwell in her anterior world of myth and legend. Still, in 1932, two years before publishing Mary Poppins, she had made the lefty intellectual's obligatory trip to Soviet Russia and concluded that "in a world rocking madly between fascism and communism" she leaned towards the latter. And you can see that, surely, in the way that Mary Poppins's magic world is peopled not by eccentric duchesses or twinkly godmothers, but by park keepers, zoo attendants, policemen, butchers, confectioners and the old woman who feeds the birds on the steps of St Paul's. It is these low-status individuals, inhabitants of the economic and social underworld, who are summoned to teach Jane and Michael their life lessons.

But just what those lessons are remains hard to say. There's certainly no nursery morality going on here, no learning to be kind to others or remembering to say "thank you". Mary Poppins rules through coldness, stares, lies and, on occasion, downright terror. When Michael is naughty she threatens to hand him over to "the policeman" and she sends grumpy Jane to exile inside a cracked Doulton bowl. Cruellest of all, she constantly threatens to leave, playing on her young charges' terror of abandonment.

In lieu of anything approaching an overarching narrative, perhaps the best way of understanding Travers's Mary Poppins is as a set of self-contained fairytales. Each chapter precipitates Jane and Michael into an adventure as terrifying as anything from the Brothers Grimm. There are children in cages and a woman who snaps off her fingers and gives them to the children to nibble. All this is made more frightening by the ordinariness of the setting and props. Borrowing perhaps from Andersen, Travers gives domestic furniture and banal objects the ability to talk and think and feel, making the everyday world seem suddenly uncanny. The lessons that the Banks children learn are not the usual ones about relating to the external world – the virtues of being kind and polite so that one day they may become fully socialised adults. Instead, these are much more profound, internal, enduring lessons about the way in which grownups, even those who are supposed to be caring for you, will always let you down, maybe even wish you harm. And that this betrayal, in the end, is survivable.

This is not to say that reading Mary Poppins now, armed with various bits of adult knowledge, is consistently thrilling. I still think Travers let her love of esoteric philosophy unbalance her book. There are chapters – when one of the Pleiades comes to earth or Mary Poppins is hailed as a goddess at the zoo – which sound as if they could have been written by the theosophist Madame Blavatsky on a particularly dotty day. In fact, it is just possible that my eight-year-old self was right, or at least not all wrong. Walt Disney took a small, difficult book – not yet a classic in the way that Winnie-the-Pooh or Peter Pan were when he got his hands on them – and he stripped it down to its component parts and reimagined. The film became the yang to the book's yin. Which is exactly the way of looking at it that would have made Travers, if not Mary Poppins herself, break into one of her very rare smiles.