A few days before setting off, we are sent our first “route card”, complete with a map, a few warnings (“there is a high risk of midges and ticks in this area”), and the promise of “a sheltered meadow-woodland space ideal for roaming”. Then comes the start of this long weekend: just after lunchtime, I set out on a three-mile walk from a nearby car park with my son James, 18, and daughter Rosa, 16, arriving at our destination in the late afternoon in searing heat. We have made sure to bring three vital litres of water. Once our tent is pitched and the evening’s relative cool arrives, what we half expect materialises: a lovely feeling of time ceasing to matter, which runs through an evening spent eating dinner (the obligatory instant pasta), aimlessly rambling around our surroundings, then marvelling at a sky much starrier than any to be seen in a town or city.

One of CampWild’s rules is that locations must be kept secret, in case word gets out and they are overrun with unauthorised campers. This much I can say: the land we are staying on is part of a regenerative farm that claims to produce 167 varieties of food, and whose proprietors are enthusiastic rewilders and tree-planters. Its co-owner, Kate Hughes, tells me she welcomes campers because “if we don’t have people on the land, they won’t fight for nature: we have to have a relationship as a nation with the natural world that supports us”.

Our somewhat limited knowledge of bird calls suggests that we are in the company of wood pigeons, one or two sparrowhawks and an abundance of blackbirds. We are half hoping to see a deer or two, but although none materialise, it hardly matters – this feels like somewhere teeming with life.

Somewhat inevitably, James and Rosa spend time on their phones, but we soon agree on a compromise: 90 minutes spent listening on a Bluetooth speaker to suitably pastoral music – Nick Drake, Fairport Convention, the acoustic demos for the Beatles’ White Album – before a final hour of stillness and silence, when we begin to drift off to sleep. James has always been much better suited to staying outdoors than in (his first recorded lie-in happened on a Dorset campsite when he was five). So it proves tonight. By 11pm, he is slumbering, while Rosa and I stay awake for another half hour.

CampWild was founded by Alex Clasper and Tom Backhouse, thirtysomething dads whose lifelong passion for the outdoor life was ignited on camping trips arranged by their Devon comprehensive school. Several years after they first met, Backhouse’s sister was involved in a serious car accident, which led him to do a sponsored trek around all of the UK’s national parks to raise money for the air ambulance service that rescued her.

Clasper accompanied him on some of these adventures, which involved a good deal of wild camping and sparked a revelation. “Escaping, getting off grid and spending time in nature was almost like therapy,” Clasper tells me, a few days before I set off. “Sitting under the stars for the evening – that’s where we’ve had some of our deepest and most important conversations.”

For many people, spending a night or two this way can seem daunting: CampWild’s essential modus operandi, Clasper says, is to “give them the confidence and knowledge and knowhow: a bit of guidance and hand-holding”. And what they offer has chimed with the zeitgeist in two ways. Over the past two and a half years, awareness of wild camping has rocketed, thanks partly to the legal tussle between the Dartmoor landowner Alexander Darwall and Right to Roam activists, which was finally settled – in the latter’s favour – by the supreme court in May.

At the same time, the collective yearning for nature, manifested in a deluge of books about hares, footpaths and rivers, has surely accelerated CampWild’s growth. It now has about 4,000 members, who pay a £25 annual fee – £1 of which goes to the environmental charity Rewilding Britain – and are charged about £15 per stay, with fees going to the landowner.

Another rule, aimed at gently enforcing meticulous standards on litter and mess, is that campers must take a before-and-after photo of their spot, and mail it to CampWild within 24 hours. But one question, Clasper tells me, always comes up: what to do about the most basic human functions? Poos must be bagged up and disposed of elsewhere: “There are a couple of spaces that do allow, er … digging, but most don’t fall into that category.” By way of highlighting roughly how to do it, CampWild has a sponsorship agreement with a brand called Dicky Bag, which offers reusable receptacles – usually marketed at dog-owners – with “odour proof seams and seals”. Free weeing, needless to say, is allowed, providing it is done well away from what Clasper calls “water sources”.

Back in our field, we wake after 7am, and slowly make our way into a morning gripped by more heat. The route back to the car, along a mixture of tree-lined roads and field paths, passes through the Somerset village of Roadwater, where we are offered a lovely kind of respite. Every other month, there is a community breakfast in the village hall, and a meal for the three of us costs little more than £20. We split the afternoon between the village of Porlock and tourist-filled Lynmouth and Lynton, before the temperature begins to ease. We then set off on a 20-minute drive along isolated Exmoor roads, during which a huge deer vaults on to the tarmac 10 metres in front of us and then disappears into the countryside beyond.

This evening’s sleeping spot is stunning. In an area reportedly popular with people walking from Land’s End to John o’Groats, it lies half a mile or so beyond a huge campsite whose residents enjoy snooker-table lawns. Our chosen spot, by contrast, is the knobbly ground in a steep-sided stretch of the Exe valley, directly under a pyramid-shaped hill. The river is right next to us: six or seven metres wide, scattered with pebbled islands. The night sky is particularly vivid: James once again falls asleep almost instantly, while Rosa and I manoeuvre our heads next to the tent door and stare up, half-convinced we might be in the presence of UFOs, before we realise they are – obviously – distant planes, presumably en route to Bristol airport.

As we drive home, I can feel the meditative calm the weekend brought me still lingering, along with the sense that this bucolic version of Airbnb is going to become even more popular. “We want to get 1 million people across the UK out into these spaces, experiencing nature and slowing down,” Clasper tells me. I slightly worry that those imagined multitudes might get in the way of all that gorgeous quiet, but it might just happen.