Grant Hazlehurst, a civil servant from Bromley, Kent has seen many whales. “Fin, sperm, Cuvier’s beaked, True’s beaked, sei, long-fin pilot …” most of them from his regular jaunts on a car ferry in the Bay of Biscay. “But I never thought I would see a beluga, not in the Thames,” he said. “So, I’m hoping.”

So were the two dozen or so others who, on Friday morning, gathered on a windy shore near Gravesend, scanning foam-flecked waves in anticipation that, for a fourth day, the beluga whale that has somehow got lost in the Thames, would show itself.

Two hours in and Bill, who had travelled from Somerset, had only managed to tick off a sparrowhawk. He had heard about the beluga on a birding app. “But I didn’t believe it. I was waiting for it to come up: ‘Correction. Albino porpoise,’” he said. Now he was here, and – ping – his alert went off: “Whale spotted, just west of the Ship and Lobster pub.” It was a couple of kilometres away. He was off at a trot. “I’m a happy man,” he said later, after managing to see it.

Peter Cartwright, was one of the first to spot it on Friday. “First you see this big melon head coming up, and then it rolls over. It’s just like a big white tyre,” said the counsellor from south-east London. He and his brother had, cunningly, split up and gone in different directions to maximise their chances. Was it exciting? “Absolutely. It’s a beluga,” he said.

Beluga whales, which can grow up to six metres (20ft) in length, are normally at home in icy Arctic waters around Svalbard, Greenland, or in the Barents Sea, so the warmer temperature of the Thames is problematic.

Rescue teams have been on standby since the whale was first seen on Tuesday. But the plan seems to be to monitor and wait as long as the animal appears healthy and is not showing distress.

The whale in the Thames on Wednesday. Photograph: Peter Nicholls/Reuters

How long a wait is not an easy question, even for the experts. Though two belugas were briefly spotted off the Northumberland coast in 2015, this week’s sighting is the furthest ever south. It appears to have a food source, but its thick blubber is designed for the Arctic.

“We’ve never had the opportunity, thankfully, to observe such a situation before, so we really don’t know how long it could remain,” said Lucy Babey, head of science and conservation at the charity Orca.

First to spot the cetacean, known to ancient mariners as the “sea canary” for its distinctive clicks, squawks and whistles, was ecologist Dave Andrews, working on a bird survey in the area on Tuesday. A white shape caught his eye but he wasn’t sure if it was just a wave. “About one hour later, I heard the whoosh sound of a blow of a whale,” he said, “and it was just out in front of me. I was quite shocked. I had seen them before in the Arctic so I knew. But I couldn’t quite believe I was seeing it there in the Thames.”

Andrews shot video and posted to the Rare Bird Alert and to Twitter. Then, he said, “the craziness started”. Within two hours TV helicopter crews were flying overhead, and around 200 people had descended on Gravesend, with others also on the opposite bank at Tilbury.

Wildlife writer and photographer James Lowen immediately sped from his Norfolk home. “It was a once-in-a-life time sighting,” said Lowen, who has visited the Arctic but never before seen a beluga. “Failed on the Svalbard, yet succeeded on the Thames. Incredible.”

“Seen with the naked eye it was was like a moon rolling through the water, but with binoculars, you could see this great melon-shaped forehead. Some people were lucky enough to see it ‘spy hopping’ which is when it pops its head above the water to look around.”

Wildlife cameraman Richard Taylor-Jones, who saw the Twitter alert, was “quite incredulous”. “But there was no doubt about it. The pictures said it. It was a beluga whale. So I jumped in the car because I thought this is an opportunity I simply cannot miss.”

He added: “Lots have come down from London. You hear people saying: ‘Crikey, I was supposed to be in the office today.’

“There are some very keen people and lots of smiles. It’s a very jovial atmosphere, because it’s something extraordinary.”

On seeing the whale, he felt “excitement, and shock, and incredulity”. “But then, very quickly followed by a degree of sadness, because I just can’t see this story ending well. I hope, of course, that it’s going to make its way back out to sea. But it just shouldn’t be here,” he said.

The whale has been spotted taking deep breaths and diving, as though feeding, and swimming vigorously against the tide. Volunteers from the British Divers Marine Life Rescue have inspected it at close quarters from a boat, and believe it appears healthy.

Brian and Maureen Philpot, who travelled from Bedfordshire on Friday. Photograph: David Levene/The Guardian

The Port of London Authority is protecting the whale by warning boats to avoid the area, and urging the public to admire it from the shore and not approach in boats, or fly drones or TV helicopters over it.

One fear is it may swim further up the Thames towards London. Martin Garside, a spokesman from the Port of London Authority, said it had remained in roughly the same spot since Tuesday. While it was a marvellous spectacle, he maintained: “The best thing for this whale is that it is never seen again.” The hopes are it will find its own way back out to sea.

Something has gone wrong with its sophisticated sonar navigation system, either through illness or through its system being confused by environmental factors, including, possibly, shipping traffic or the construction of platforms, experts believe.

Said Babey: “It’s a little bit sad to see it, mixed still with that excitement that there is a beluga in our waters. But the overriding concern is it is thousands of miles away from its normal home. So, it does raise the questions why it’s here, and will it be OK.”

While it remains here, however, the whale continues to draw spectators. Brian Philpot, and his wife, Maureen, set up camp on Friday at 8am, sheltered from the wind by a bush, having travelled from Dunstable, Bedfordshire. “We are really hoping to see it,” said Brian, an electrician and self-confessed “twitcher”. But, he conceded, it probably would not compete with the black-browed albatross he once spotted off Scotland.