“Green surge”: these are two words Green party supporters are beginning to say with no little relish. Britain’s old political model – a future miraculous revival notwithstanding – appears to be disintegrating in the face of a multi-pronged assault by the SNP, Ukip and, increasingly, the Greens.

The last of these really believes that its time has come. At the 2010 general election, despite the crusading Caroline Lucas pulling off a historic victory in the tight three-way marginal of Brighton Pavilion, the Greens did poorly. Barely registering a single percentage point, they actually chalked up a smaller proportion of the vote than they had in 2005. Now look at them: beating the Lib Dems in the European elections, usurping Nick Clegg’s party in two opinion polls, soaring party membership. The party’s leader, Natalie Bennett, cheerfully talks of a “peaceful revolution” in British politics – hyperbole, perhaps, but who knows?

The media treatment of the Greens is beginning to look painfully absurd. They had an elected MP years before Ukip appropriated Douglas Carswell from the Tory backbenches; they run a council; they’ve long boasted a presence in the European parliament. And yet the likes of the BBC all too often prefer to act as Ukip’s unofficial campaign team. Remarkable, perhaps, that the Greens are doing quite so well given the relative dearth of airtime.

A rich ex-Tory, ex-City type peddling Thatcherite economics wrapped in a rich coating of divide-and-rule xenophobia? You might as well hire a TV agent. Proffering a living wage, taxes on the booming wealthy, and solutions to the looming environmental calamity that create jobs? Don’t waste the afternoon expecting a TV producer to call.

“Enormous frustration and anger” is how Lucas sums up party feelings about media treatment. An increasingly common view, it seems: the BBC’s refusal to include the party in the 2015 general election leaders’ debate has provoked nearly 200,000 people to sign a petition urging a U-turn, and 47% of Britons support such a move. The Greens’ ceiling of support is potentially high indeed: according to Ipsos Mori, 43% of Britons would consider voting for them, nine percentage points above Ukip and only one less than the Tories.

A bit of self-criticism: I’m part of the British media, and have hardly been generous in column inches devoted to the Greens. Frustrated party members often berate me for backing policies on social justice, tax justice, public ownership and workers’ rights that they loudly champion, all the while trying to lobby a Labour leadership that hardly seems receptive.

The reasons for this division on the left are straightforward: many of those who resist the Greens’ flirtations have a commitment to the labour movement, to working people collectively organising for change, and an understanding that – as things stand – a general election represents a choice between a Labour-led and Conservative-led government. But media commentators of all stripes can hardly go on sticking fingers in their ears and yelping “la la la” whenever the Greens are mentioned.

An electoral system that represents a formidable block to new emergent parties aside, the Greens – increasingly giddy with success though they are – have to address a number of problems if they really are going to transform politics. Look at recent YouGov polling: while Ukip is doing particularly well among certain older working-class voters, the Greens disproportionately attract younger, university-educated middle-class types. Lucas blames a “stereotype of the Greens we’re still fighting to overcome”. Let’s just be brutally honest about that stereotype: an eccentric bohemian hippy, unkempt beard, John Lennon-style glasses, wading through muesli in dishevelled sandals. A bit like the old Lib Dems, perhaps: and indeed the Greens owe a big chunk of their surge to the exodus of voters from Clegg’s discredited rump.

Their name defines them by the issue of the environment: today’s warning from the UN about climate change inflicting “severe, widespread, and irreversible impacts” should give such a cause renewed urgency. But, however wrongly, climate change often seems too abstract, too long term an issue, particularly if you face more immediate concerns about housing, jobs and wages. The Greens are doing their best to counteract such an image. They link looming environmental disaster to the creation of hundreds of thousands of renewable energy jobs – surely crucial in a post-industrial age where skilled middle-income jobs have been stripped away by policies of economic vandalism.

After internal focus groups suggested that voters had got the message about the environmental platform, and wanted to hear what else they stood for, there has been a clear shift in emphasis. The party is talking far more about policies like a living wage, tackling the yawning gap between the highest and lowest paid, opposing austerity, building new council homes: all geared at winning working-class support.

There are other hiccups. Green-led Brighton council’s attempt to overhaul council workers’ allowances – which provoked a bitter strike after some bin collectors faced pay cuts of up to £4,000 – inflicted lasting damage on relations with unions. But their leader is keen to emphasise her support for the struggles of trade unionists, like care workers in Doncaster and NHS workers last month. Much of the labour movement is exasperated by a sense of directionless timidity from the Labour leadership: whether this delivers substantial shifts in allegiance to the Greens has yet to be answered.

Then there’s their attitude to Labour parliamentary candidates who fundamentally share Green values. The Greens stood candidates against staunchly anti-war, pro-social justice Labour MPs like John McDonnell and Jeremy Corbyn in 2010; they’re fielding a candidate against another progressive Labour candidate, Clive Lewis, in Norwich South. Emphasising that it’s her personal view, Caroline Lucas says she’s sympathetic to the idea of non-aggression pacts with such candidates; Natalie Bennett is far less so, wondering why such candidates choose to stand under a Labour banner. But such an attitude risks alienating those who fear that, however well-meaning the Greens may be, their function could be to split the progressive vote and deliver election victories for the coalition parties.

Farage’s millionaire-bankrolled “people’s army” has already turned British politics into a bidding war over who can kick immigrants hardest. Can the Greens at least trigger a similar shift in political debate on social and economic justice? As the old political system squirms, like Gulliver tied up by Lilliputians, this is one of many questions to be answered. But the Greens have earned reasonable media coverage, and it would be a democratic outrage to resist granting them it.