The mother of a seven-week-old baby boy who died while in the care of a privatised out-of-hours GP service in London vowed to campaign against commercialisation of the NHS to ensure no other family experienced the same trauma – a day after an inquest concluded that no individual or overall system failing was to blame.

Axel Peanberg King died aged just 54 days last November when a routine cold that had developed into pneumonia went untreated despite repeated calls and visits over the course of five days to the out-of-hours doctors' service run by private contractor Harmoni.

"Having heard all the evidence in court, I don't think anyone can have full confidence in Harmoni's service. If this can happen to a strong healthy baby like Axel, and a confident second-time mother, it can happen to anyone," his mother, Linda Peanberg King, said.

The days since his death waiting for the inquest – "114 days and nights to go over every event and conversation" – had been gruelling, she said. Having exclusively breastfed Axel and carried him around most of the time in a baby sling on her front, there had been the shock of feeling suddenly "physically incomplete" as well as the mental anguish of rehearsing a thousand "what ifs".

"I wake up with my heart pounding. There is a physical pain, like a spear that has gone through my heart." Peanberg King said. "CS Lewis said 'no one ever told me that grief felt so like fear' and that sense of fear never leaves me – you see danger everywhere."

She has a wooden box, given to her by other mothers in her National Childbirth Trust group when he died, in which to store his precious things, the letters about his death, his tiny pink socks – her mother thought he would be a girl. And she asked for a lock of his hair, a golden tuft nestling in a tiny carton, but she finds it hard to bear the thought of someone who never knew him cutting it from him.

But trying to make sure no one else suffers the same experience has become a way of channelling that grief for both her, a Swedish design manager for a charity, and her husband, Alistair King, a marketing manager for a legal publisher.

The couple are now working with the activist group 38 Degrees, which uses online petitions to mobilise protests. The group, which is campaigning against privatisation in the NHS, launched a petition calling for the parliamentary public accounts committee to investigate out-of-hours care outsourced to private providers when the Guardian first wrote about Axel's case last December. It had more than 130,000 signatures within 48 hours of its launch. Peanberg King was already a member before Axel's death.

The coroner, Dr Shirley Radcliffe, had concluded on Thursday that Harmoni's overall systems were safe and rigorous. She also concluded that the family's first consultations and calls to the service had been dealt with appropriately and in a timely fashion.

However she also said that the performance of the last doctor from the service in contact with the family had been inadequate. He had given just one minute to assessing Axel in a phone call, had recorded "wholly inadequate" notes of his consultation as he assessed Axel in between seeing a queue of patients, booked at a rate of five an hour, at another Harmoni out-of-hours clinic where he was on duty. By downgrading Axel's case from urgent to routine, there was a delay in him seeing a doctor on the day he died.

Another Harmoni doctor who had seen Axel two days before his death was found by the coroner to have conducted a full examination that was "appropriate" but gave a deferred prescription for antibiotics that the coroner recorded was not good practice, since it might have led to confusion over what should happen if he did not get better.

The family feel there are still many unanswered questions about how the service functioned and whether the fact that it was run for profit was a contributory factor. "I have reservations about the way some of the doctors who treated us performed, but the root causes of what went wrong are way above them. I know the NHS is hugely put upon by rising population, but the answer is not to apply normal commercial practice to the health service, so that people are looking at the bottom line," Axel's father said.

Peanberg King said that they knew people spent much time "obsessing about the postcode lottery for schools", but never expected to find the same might be true for health and the treatment your baby received. "It looks as though the whole of the UK is becoming a lottery for health."

Talking on Friday – a day after the St Pancras coroner recorded a narrative verdict, which does not apportion blame – the couple said they both knew that families were often left frustrated by inquests. They were nevertheless unprepared for how brutal the British adversarial system of justice could feel, a system that can put a young mother, within weeks of bereavement, in the witness box for cross-examination, and in which barristers for the parties suspected of failings are supposed to defend their clients by accusing the mother in turn.

"The whole onus is on the family to provide expert evidence and prove what went wrong," Peanberg King said. They only received a postmortem report at the end of the first week of January and learned then that pneumonia was the cause of death rather than any congenital defect or rare untreatable illness. It is very rare, although not unprecedented, for young babies to die of pneumonia in this country. They could not face postponing the inquest date to find more experts, however. She dreaded being questioned in court and wanted the ordeal over.

Ellen Parry, a solicitor at Leigh Day, who is representing Axel's parents, explained: "The obligation to investigate the cause of death rests with the state. However, the question frequently arises whether independent expert evidence should be called and who should commission and fund it. The coroner can only make her decision on the evidence in front of her and the only evidence which went to the safety of Harmoni's systems came from Harmoni doctors themselves."

The inquest also required the family to relive Axel's last week of life in harrowing detail. By the time Peanberg King arrived at the Harmoni clinic on 3 November, her baby was already dying, although she did not know it. She was asked to wait in a queue with six patients in front of her.

It was an off-duty paediatric nurse from the NHS A&E next door who realised it was an emergency and ran with mother and baby into A&E shouting frantically for the resuscitation team. Overwhelmed by the bacterial infection in his lungs, Axel had gone into cardiac arrest, but his father was at home, caring for their other son Carl. When Peanberg King told her husband that their baby was dying, he was so stunned, he started screaming. "I couldn't function," he recalled; he was unable even to call a taxi. A police car was dispatched to bring him and Carl to hospital. The emergency team kept up their unsuccessful efforts to save Axel long enough to allow them to arrive to say goodbye. The last morning of the inquest was largely taken up with counsel for Harmoni taking the company's medical director through the steps it had since taken. "It felt like a powerpoint exercise; it was shocking to hear your dead son being described in corporate speak as "a learning exercise'", Peanberg King said.

Socially considerate, the couple had not wanted to use A&E inappropriately when Axel became ill, knowing how stretched it can be. Axel had not fed or produced properly wet nappies for several days but they were reassured by the Harmoni doctor on 1 November, so afterwards they worried that they were being over-anxious. "As a family we thought we were accessing health services through the appropriate route and that they would know when to refer to A&E. "

Peanberg King has tried to explain Axel's death to Carl, who will be three in June, by reading him children's books with titles such as "I Miss You" and "What Does Dead Mean?" She said: "I need to hear it in these simplistic terms too, because I don't understand it either. People talk about grief in stages that lead to acceptance, but you are never going to accept that your child has died. The world has changed. You can only adapt not accept. I struggle to find meaning in a lot of things." Carl's young child's spontaneous expression of confusion often reflects their own. He told them recently: "When Axel has finished dying, he can come home."

"And you have to say, he can't," Peanberg King said.

Backstory

Harmoni is the largest private provider of out of hours GP services in the country and has been the most successful private bidder for the new government contracts to run the NHS 111 phone line for non-urgent care.

Set up in 1996 as a GP co-operative, it grew rapidly from 2004 when changes to doctors' contracts introduced by Labour allowed them to opt out of out of hours care. In 2005 it became a joint venture between the original doctors and private equity group ECI. They cashed in on their investment and the value of their NHS contracts last November when Harmoni was sold to another private equity-backed health compnay, Care UK, for £48m. Five family doctors became millionaires from the sale.

The Guardian reported last year that local doctors and whistleblowers at the Harmoni out of hours service had questioned staffing levels and feared delays in treating baby Axel may have contributed to his tragic death.

The coroner concluded this week that Harmoni had safe and rigorous systems in place and staffing levels that were optimal but found the performance of the last doctor from the service in contact with the baby "inadequate". Harmoni said in a statement after the inquest: "We believe we have the right underlying systems, policies and procedures to ensure a safe and robust out of hours service. We know that the review of very difficult incidents such as this always identifiy learning points. Our overriding priority is to ensure that this learning is acted on."