The must-have accessory for every English country estate was once a gothic folly, a ha-ha or a croquet lawn. Now it is a pair of beavers.

Landowners and large estates are racing to acquire licences to reintroduce the water-loving rodents, which were hunted to extinction in Britain 400 years ago.

Natural England has issued 13 beaver licences since 2017, and beavers are breeding and roaming as wild as they can in large fenced “trial” enclosures in North Yorkshire, Cornwall, Essex, Devon, Somerset and Gloucestershire. This year, beaver pairs will be returned to Cumbria, Norfolk, West Sussex and Dorset.

This week, a pair of beavers were released onto pools above an old watermill on the 12,500 acre Holnicote estate on Exmoor, the first National Trust property to take them. Approval has just been granted for beavers to be added to the Knepp estate in West Sussex, the former dairy farm that has been rewilded by Charlie Burrell and Isabella Tree.

But beaver-loving landowners face delays with a waiting list to acquire the herbivorous animals, which are now sourced from the wild population in Scotland, where the beaver was officially recognised as a native species once again in 2016.

“We are worried about how we will get hold of them because there are so many people who want them, which is incredible,” said Isabella Tree, whose book, Wilding, has driven a surge in interest in rewilding.

A YouGov poll this week found overwhelming public support for reintroducing beavers into Britain, with 76% of people supporting the idea, by far the most popular mammal for reintroductions ahead of the wild cat, wolf and lynx.

One ecologist described showing a group of wealthy landowners around one new beaver project. He said: “They were a shooting and fishing fraternity, made their money out of banking. By the end of the day, they all wanted beavers.”

The beaver has yet to be officially recognised – and given legal protection – as a native species in England, but new research revealing the beneficial impact of their dam-building is currently being considered by the Department for the Environment, Farming and Rural Affairs (Defra).

Beavers can be secretive and nocturnal but they are also placid, slow-moving animals, which can be watched swimming in summertime. Photograph: Cain Scrimgeour/PA

While licenses only permit beavers to be placed in large “trial” enclosures, escapees from privately owned collections have created a burgeoning population of truly wild beavers on the River Otter in east Devon.

The Guardian has learned that there are other secretive populations of free-living beavers on rivers in Kent, north Somerset, Gloucestershire, west Devon and Wales.

Beaver ecologist Derek Gow, who has overseen many of the licensed reintroductions, said escapes occurred because beavers’ strong teeth were like bolt-cutters.

Hailing beavers as “ecosystem engineers”, Gow said: “They turn landscapes that are largely dead into environments that wildlife can recolonise.”

The five-year government study of both fenced and wild beavers will show how their dams can prevent flooding by drastically slowing the flow of streams, purify water polluted by nitrates and phosphates, store carbon and create an explosion in other wildlife.

“All of these benefits could accrue in every single headwater stream or catchment in England,” said Prof Richard Brazier of the University of Exeter, who led the study. Brazier’s modelling – currently being examined by Defra – identifies suitable habitat where beavers could improve river catchments in virtually every part of England, Scotland and Wales.

During five years monitoring a two-hectare study enclosure in west Devon, Brazier and colleagues found that frogspawn increased from 10 clumps to 650 clumps last summer, with amphibian life attracting predators such as herons and egrets. Dragonflies, butterflies, aquatic plants, flowers and trees have benefited from new pools created by the beavers. “Bringing that type of biodiversity back is something that you couldn’t engineer,” said Brazier.

In Cornwall, Somerset and the Forest of Dean, new beaver enclosures are expected to reduce flash-flooding in villages downstream.

The beaver has friends in high places: both the environment minister Zac Goldsmith and former environment secretary Michael Gove have become beaver converts, and the prime minister, Boris Johnson, has spoken in favour of the animals.

Some landowners are won over by potential economic benefits such as ecotourism. Beavers can be secretive and nocturnal but they are also placid, slow-moving animals, which can be watched swimming with their kits in summertime.

But not everyone loves the beaver. While beavers will not dam large rivers because the waters are already deep enough for them to feel secure, their dam-building can cause flooding on flood-plains in lowland areas. In Scotland, farmers have shot beavers blamed for flooding farmland.

The government is expected to announce later this year that the free-living River Otter beavers will be allowed to stay permanently, paving the way for official, licensed releases of free-living beavers beyond enclosures.

Gow said: “We need to move beyond just keeping them in pens. In Britain, species recovery has always been about wee spiders or dormice – things that don’t change landscapes. We have to learn to tolerate animals that change things for the better.”

A Defra spokesperson said: “We are committed to reintroducing formerly native species, such as beavers, where there are clear environmental and socio-economic benefits.

“Beavers have a special place in English heritage and can play a role in creating new havens for wildlife and boosting our natural biodiversity. We welcome applications for trial reintroductions of beavers, where these benefits are clear.”