Marieme and Ndeye each have a sticker on their faces: a butterfly for Ndeye, and a green smiley face for her twin sister. They giggle as they take them off and stick them back on; then Ndeye decides it’s their dad’s turn, placing the smiley face over his right eye.

“Ndeye is the lively one, she likes attention, and Marieme is a quieter personality – calm and thoughtful,” said Ibrahima Ndiaye, the twins’ father. “Ndeye is fire and Marieme is ice.”

Their behaviour – and their differences – are typical for three-year-old twins, but Marieme and Ndeye are not typical at all. The sisters are conjoined: they have separate brains, hearts and lungs, but share a liver, bladder and digestive system, and have three kidneys between them.

Ndiaye brought his daughters from Senegal to Great Ormond Street hospital (GOSH) in London at the age of eight months after a desperate search for medical help. Over the past two and a half years, he and the hospital have wrestled with an agonising decision about whether to go ahead with a surgical separation that Marieme would not survive, but that could give Ndeye a chance of a reasonable life. Without a separation, both will almost certainly die.

The dilemma is the focus of a BBC documentary, The Conjoined Twins: An Impossible Decision, to be broadcast on Monday. It follows the deliberations of the hospital ethics committee, during which clinical and lay members, along with Ndiaye, navigate existential questions presented by scientific and medical advances.

“Decisions are much more complex than they used to be,” said Joe Brierley, a consultant paediatrician and chair of the ethics committee. “We can do unbelievable things compared to 20 or 30 years ago. But just because we can, it doesn’t always mean we should.”

The committee’s role was not to make decisions but to steer clinical teams and families through “tough moral dilemmas, and ensure different points of views and values are aired”, he said. The ethics committee was the first in the UK to invite patients and families to take part in discussions – sometimes an uncomfortable process, said Brierley.

Marieme and Ndeye were born in Dakar in May 2016. Ndiaye, who has four older children, paid for four separate scans during his wife’s pregnancy. None even indicated twins, let alone conjoined twins. So their birth was a “massive shock”.

In the following months, he contacted hospitals all over the world, asking if they could offer help. Each time the blunt answer was “no” – until Great Ormond Street said, “Come and we’ll see what we can do”. The hospital has separated more than 30 sets of conjoined twins, including this year Safa and Marwa Ullah from Pakistan, who were joined at the head.

It was a light in the darkness, said Ndiaye. “I came [to London] with a lot of hope. However difficult the situation, I told myself, I’m in the UK and they will find a solution.”

The family arrived in January 2017, when the twins were eight months old. The medical team, led by Professor Paolo De Coppi, quickly established that Marieme’s heart was dangerously weak and her oxygen saturation levels low. “Paolo told me we can’t do [the separation] without losing Marieme. The light, the hope, the expectation – all of a sudden, this vanished,” said Ndiaye.

He faced an agonising decision: should he give his permission for surgery, knowing that Marieme would die, in order to give Ndeye a chance of life? Deciding against surgery would almost certainly mean Marieme’s health would deteriorate and both girls would die. But Ndiaye simply could not contemplate knowingly causing Marieme’s death.

“My emotional link with the girls was so strong, I was very attached to them. It was a very difficult moment,” he said. “In this situation, you don’t use your brain, you follow your heart. Any decision is heartbreaking, so much turmoil, so many consequences.”

There was a legal precedent. In 2000, the UK high court ruled that Maltese conjoined twins “Mary” and “Jodie” should be separated against their parents’ wishes. Without surgery, both would die; with surgery, Mary would inevitably die but Jodie could have the chance of a full life. It was an “excruciating dilemma”, said one of the judges.

The case was brought by St Mary’s hospital in Manchester, where the twins were born and cared for. The girls’ parents, devout Catholics, argued that surgery was “not God’s will”. The surgery went ahead; Mary died, Jodie survived.

At an ethics committee meeting to discuss Marieme and Ndeye, Brierley raised the question of whether it would be right to go to court if the twins’ father and the clinical team came to different conclusions. The documentary shows him gently spelling out to Ndiaye the consequences of not separating the girls: “Marieme’s dying process will be Ndeye’s dying process – it isn’t possible to stop that or change it … [And] it won’t be an option to separate them once Marieme starts to die.” In the end, he said, “there was no disagreement” with Ndiaye’s painful decision not to separate the girls.

It was the only possible conclusion, Ndiaye told the Observer: “They are together , they are equal. Great Ormond Street had been very honest and very clear with me all the way. We came [to the hospital] as patients, but now we are more than that. I consider [the team] as family. I have never felt under pressure to agree to an operation. I have never been disrespected.”

Brierley described Ndiaye as an “incredibly dignified, thoughtful, eloquent man and a wonderful father. You see those girls cherished, and they know they are cherished. It’s an impossible situation for everyone involved, but he will live with his decision for the rest of his life.”

Ndiaye has been sustained by his faith as a Sufi Muslim. The Qur’an tells him to be strong, honourable, dignified and patient in the face of hardship, he said. His wife has returned to Dakar and he has lost his job as a project manager in tourism and events. After granting him discretionary leave to stay in the UK, the Home Office moved Ndiaye and the girls to Cardiff. “I have put everything on standby. I put all my energy into caring for the girls and making them happy.

“I know there will be a time when they have to go. But at this point they are fighting – and they are also providing me with a reason for living. They are my inspiration, I dedicate everything to them. I will never let them walk alone,” he said.

“I need to know I gave them everything I could. I’m a lucky man to be part of this journey. We are still on the journey. I don’t know how it will end.”

The Conjoined Twins: An Impossible Decision is on BBC2 on 5 August at 9pm