If news reflected the magnitude of the planet’s crisis, one story would splash the headlines every day, filling every column inch, airtime minute and swipe of TikTok. Writing about anything else often seems trivial. Cop26 will briefly ignite the news, but for how long afterwards? Be grateful to Extinction Rebellion and Insulate Britain activists, who don’t give up, though they too need to keep finding new ways to capture public attention.

With the climate summit less than a fortnight away, the government’s inadequate “plan” for net zero shines a flickering light on how much more needs to be done. Cop26 risks exposing the weak will of the world’s leaders to take action. Far from holding to the vital 1.5C heating limit, the UN says cuts volunteered by countries so far will heat the world by 3C above pre-industrial levels: certain calamity. Commitments still fall 60% short of the 2050 net-zero target, needing $4tn investment this decade alone.

As the first major economy to set legal goals for net-zero emissions by 2050, Britain is good at targets. But the government plans fail to spell out how we’ll get there, at what price, and who will pay. Embarrassments include a proposed deep coalmine in Cumbria, neatly hidden by an “inquiry” that will not report until after the climate summit. Nimby planning authorities block over half the applications for the cheapest renewable onshore wind, despite strong public support. Without those wind farms, RenewableUK says, Britain won’t be able to decarbonise its electricity by 2035 as pledged. Home insulation, the most practical, lo-tech action, doesn’t feature; nor does cutting back on meat and dairy.

There’s a danger of retreating into nihilism. Turn the page, think about something else, and don’t look your grandchildren in the eye. But one event at Cop26 will raise the spirits. The NHS, far out ahead, will make a key presentation to the world’s other health systems to show what can be done. The NHS is responsible for 5% of the UK’s carbon emissions: the recently departed head of NHS England, Simon Stevens, put climate targets firmly into the NHS long-term plan, warning: “If health services across the world were their own country, they’d be the fifth-largest emitter on the planet.”

You might think going green was the last thing the NHS could do right now, after its decade of unprecedented underfunding, with waiting lists going through the roof, acute doctor and nurse shortages, and burnt-out staff being awarded paltry pay rises. But no. Jackie Daniel, head of Newcastle upon Tyne hospitals, has pioneered the green cause within the NHS and is making the Cop26 presentation to world health services. “A few, exhausted trusts did think this was the last thing they could cope with now,” she tells me. “But astonishingly, nearly everywhere has seized on this. We’ve found it’s something that really enthuses staff, and they have been the ones driving it forward.”

From clinicians and technicians to cleaners and chefs, they have taken the initiative with their own climate-saving plans – from vegan options to meat-free Mondays in canteens, to the first baby born with a gas-and-air system that captures the polluting gas instead of emitting it.

Remote consultations – oddly under attack by the health secretary, Sajid Javid – are a huge contributor to cutting journeys: the NHS saved 14m miles by using remote consultations last year, Dr Nick Watts, NHS England chief sustainability officer, tells me. These helped cut Newcastle’s travel emissions by 43%, along with walking, cycling and electric buses for staff while at work.

Anaesthetics turn out to be climate killers: one vial of desflurane, the commonest, has a carbon footprint equal to the emissions from 350 kilos of coal. Switching to sevoflurane is as effective and less destructive, according to Watts. The climate crisis is a health crisis, he says, sometimes in very direct ways: air pollution kills up to 36,000 Britons a year, and hospitalises many more. He aims to cut 3% of NHS emissions this year – which he says is the equivalent of 1.7m flights London to New York. But, in a wicked spiral, the asthma inhalers necessitated by the UK’s pollution, using hydrofluoroalkane as a propellant, themselves release 4% of NHS emissions. Switching just 10% of them to dry powder inhalers has so far cut CO₂ emissions equal to taking 120,000 cars off the road. Meanwhile, 18m patients attend GP clinics in streets exceeding World Heath Organization air pollution limits.

The NHS long-term plan pledges net zero by 2040. Ideas big and small need to be spread to the world: the Royal Cornwall hospital has devised a way to recycle PPE, melting its plastic into litter-pickers. Watts says changing to LED lighting repays its cost within three years. The long list of NHS initiatives from around the country is remarkable and heartening. At a time like this, how do the staff and managers summon the energy and determination? It’s a welcome reminder that the NHS is the best of us.

We may be weary of Boris Johnson’s vainglorious boasting about “world-beating” ambitions. But the NHS will be on parade as genuinely well ahead in how to green health services, an initiative that sprang from within and has been taken up and spread by its own enthusiastic staff.

But here is the sobering truth that reflects every aspect of the climate catastrophe confronted at Cop26: good though this progress is, the health service still has vast, as yet unfunded capital costs to reach net zero. Newcastle hospitals can’t make much more progress while aged and leaky estates need massive investment to rebuild, to insulate and replace gas boilers with heat pumps and solar panels – that’s where 90% of the remaining emissions are. Daniel says: “That work must start now. We need a nationally funded plan to decarbonise our heat and supply chains. There is no human health without planetary health.”

But a leak to the Health Service Journal revealed how the government’s promised “40 new hospitals” programme is being downgraded and redescribed. At next week’s budget, there is no sign yet that the chancellor plans the kind of enormous capital spending it would take to green the NHS estate – let alone the whole country.

• Polly Toynbee is a Guardian columnist