Take one voguish designer, one national treasure and one icon-hungry mayor and what do you get? A floating forest across the Thames. But can anyone actually say what the £175m garden bridge is for?

The river Thames has long inspired a kind of architectural madness, the looping grey serpent driving many to attempt to leap its great breadth in ever more elaborate ways. As a student, John Soane proposed a triumphal bridge, a classical palace on piers that would have spanned the river with a domed temple, flanked by an avenue of corinthian colonnades. It won him the Royal Academy gold medal in 1776, but the plans have remained in a drawer ever since.

In the 1960s, a radical group by the name of the Glass Age Development Committee dreamed up a multi-storey pleasure bridge for Vauxhall. This would have straddled the Thames with a vertical stack of roadways, shops, skating rinks and a hotel, all topped with a roof garden and open-air theatre, but it proved one megastructure too far.

Not to be deterred, the Royal Academy organised a competition for a "living bridge" in 1996, won by the French architect Antoine Grumbach with a lavish suspended garden, lined with hedges, trees and an exotic "topiary cafe". It was to hang from two vast apartment towers that would have paid for the project – but these generated fierce opposition that revealed the scheme as nothing more than green garnish for a lucrative private development.

Now there is another garden bridge plan. Bursting out of the river in the form of two conjoined mushrooms, it would create a floating forest between Temple and the South Bank, held aloft on a shimmering copper canopy. It is scarcely less improbable than the heroic failures that have gone before – and yet it seems very likely to happen. It has garnered not only the support of London's mayor-cum-novelty-infrastructure-tsar, Boris Johnson, who has pledged £30m from his transport budget, but also the backing of central government, in the form of a further £30m from the Treasury. A detailed planning application has now been submitted, with the aim of having it built by 2018.

The potential view from the Garden bridge. Photograph: Arup

"It feels like we're trying to pull off a big crime," says its designer, Thomas Heatherwick, with a twinkle in his eye. He is sitting in his Kings Cross studio in front of a 2m-long model of the bridge. Beautifully assembled in timber, it depicts the two toadstools rising from their slender stalks to form a majestic pair of fan vaults above the Thames. It is at once gothic and futuristic, slick and organic, a structure that, like many of his craft-tech concoctions, wouldn't seem out of place in Middle Earth. And atop the fungal forms, in a touch of fairytale sorcery, sprouts a forest.

"It's basically just two big planters," says Heatherwick, describing the design evolution as a form of "guerilla gardening", as if two seed-bombs had been hurled into the river and sprouted up to make the bridge. Above each sinuous planter rises a thicket of trees – which will grow to a maximum height of 15 metres – thinning out towards the middle of the bridge, where the two mushrooms meet at a slender point, to retain strategic views along the river to St Paul's. Designed by Dan Pearson, the garden will take visitors on a journey from pioneer species, through windswept scarp, to a "cultivated glade" along meandering brick pathways, with a wild feeling similar to New York's High Line. "It's not just a bridge with green sideburns," says Heatherwick. "It's a proper garden. It has the potential to be the slowest way to cross the river, with intimate moments and a lingering scale." He pulls out the famous image of Leonardo DiCaprio and Kate Winslet perched at the prow of the Titanic. "It was the biggest ship in the world, but it also had this intimate point," he says. "I think public infrastructure tends to be far too big. We're interested in how big can also have small in it." Accordingly, the bridge's radial ribs will define a staggered, serrated edge to the deck, forming lots of little prows – 88 in all – for multiple Kate-and-Leo moments. If any bridge was designed to attract love-padlocks and proposals, then this is it.

If the design was inspired by guerilla gardening, it is guerilla planning that has brought this fantastical project quite so far. It began as a dreamy idea in the mind of actor-campaigner Joanna Lumley back in 1998, as a memorial garden bridge for Princess Diana. The people's princess has since given her name to a slab-and-girder crossing in Stockton-on-Tees, while Lumley's idea was subject to a brilliant parody by mischievous architects FAT, who conceived the memorial bridge as a rolling slice of Althorp park (the country seat of the Spencer family), with the lyrics of Candle in the Wind etched along its balustrades.

Fifteen years later, in the heady haze of post-Olympic self-congratulation, the idea was revived as Lumley joined forces with Heatherwick, newly validated by his dramatic cauldron design. Boris took note and duly had Transport for London invite a small number of other practices (their names undisclosed) to tender for the project. The winner was a foregone conclusion.

"We didn't want any old bridge," says Richard De Cani, Transport for London's director of strategy and policy, who was instrumental in the Emirates Air Line cable car, east London's mostly empty aerial sponsorship opportunity. "We've got our lean, mean, efficient footbridges, like the Millennium and Hungerford, so we were interested in a bridge that did something else."

He describes the £175m project as "supporting economic growth and development", bringing some of the South Bank's bustle to the "dead world" of the north bank. Lifting off from just east of the National Theatre, the Garden bridge will cross the river, and the roaring dual carriageway of Victoria Embankment, to land on top of Temple tube station, one of the least-used stops in central London. The area has been newly christened Northbank, and there are plans to pedestrianise the stretch of the Strand between the Aldwych crescent, a scheme into which the bridge neatly dovetails as a benevolent bringer of crowds.

Presented by the charismatic duo of Lumley and Heatherwick and backed by the heavyweight board of the Garden Bridge Trust, it makes a seductive fairytale.

"It will be like a tiara on the head of our fabulous city," coos Lumley. "It will set hearts racing and calm troubled minds. It will enchant everyone who uses it."

This may well be so, but there is a niggling feeling that, like many of Heatherwick's inventions, it is a spectacular solution to a problem that doesn't really exist. In his book Making, his projects are presented as answers to questions, such as: Can you squeeze a chair out of a machine, the way you squeeze toothpaste out of a tube? Can a Christmas card be bigger than its envelope? His answers never fail to bring a smile or a gawp with their formal ingenuity, but there is a sense that he is inventing these challenges in order to solve them with impressive gymnastic feats. Like his new distillery in Hampshire, the Garden bridge piers could be squeezed out of the same magic tube of architectural icing.

'It's not just a bridge with green sideburns' … Thomas Heatherwick. Photograph: Nick Cunard/Rex

Questions it might be useful for Heatherwick and Lumley to turn their energies towards include: how can the long-delayed plan for a Thames crossing in east London be revived? There are 16 road bridges west of Tower Bridge, but none to the east, where the city's growth is burgeoning, until you hit the far-away Dartford crossing. Or how about: what's happened to the long-awaited plan for a pedestrian bridge at Nine Elms, where 20,000 new homes are being built? De Carni says TfL is "scoping options" for both of these projects, but he is reluctant to be drawn on why the Garden bridge – just 300 metres from an existing crossing – has been so magically fast-tracked and received such substantial public funding. Or why a well-worked through scheme to make Victoria Embankment a pedestrian-friendly experience has simply fallen by the wayside. Even Johnson himself has admitted that he's "not really sure what [the Garden bridge] is for," other than making "a wonderful environment for a crafty cigarette or a romantic assignation".

When it is built, these minor points of reason and reality will be long-forgotten, trampled beneath the crowds of enraptured tourists. And flock to it they must, for there is something uniquely English about the fact that a delightfully superfluous piece of public infrastructure can appear at the whim of a celebrity and the impulse of an icon-hungry mayor. We are, after all, a fine nation of folly-builders.

• Thomas Heatherwick, Dan Pearson and Joanna Lumley will discuss the Garden bridge project on Thursday at the Siobhan Davies Dance Studios, London SE1, as part of the London Festival of Architecture. Details: londonarchitecturediary.com