It’s Saturday morning and Edith, a 60-year-old lady from East Lothian, has run out of the blood pressure medication atenolol. In the past she might have crossed her fingers and waited until Monday. Today she picks up the phone and dials 08454 24 24 24.

Within a couple seconds her call has been answered by Rea Thomson, a microbiology student who works part-time as a call handler in NHS 24’s centre at South Queensferry. Thomson, one of 102 call handlers on duty, deftly amasses Edith’s personal details, types them into the computer, listens sympathetically to her explanation that “the pink pills and the blue pills all look the same” and resists the urge to roll her eyes heavenwards.

A few seconds later and she has put the call through to a nurse who double-checks the details and links Edith to a pharmacist. A prescription will be waiting at her local chemist within an hour or so. Edith, who sounds like an NHS 24 regular, wonders whether “just this once” the prescription might be delivered to her at home. Thomson is doubtful, but you can see why Edith asks. A service this slick might conceivably gift-wrap the medication and courier it round in a stretch limo.

Edith is one of a growing number of patients who have stumbled upon the best-kept secret of our overstretched health service. Since 2004, NHS 24 has been the first port of call for all non-emergency, out-of-hours medical cover in Scotland. It handles in excess of 120,000 calls a month. But after a series of disasters, its name has become synonymous with delays and poor advice.

A high-profile fatal accident inquiry in Aberdeen into the deaths of 17-year-old Shomi Miah and 30-year-old Steven Wiseman, who were advised by NHS 24 to take painkillers despite having fatal conditions, has ensured a continuing barrage of bad publicity. But the service I witness bears about as much relation to the service I have read about as Chantelle Houghton, the Celebrity Big Brother star, does to Dame Kiri Te Kanawa.

Saturday morning is one of the busiest times for NHS 24. In one of the two large open-plan rooms, banks of uniformed nurses sit at work stations, their traditional thermometers replaced by telephonists’ headsets. Here Philippe Hourcastagne, a French contact centre specialist, is sitting before a computer screen covered with constantly changing numbers, monitoring the calls coming in to the eight centres throughout Scotland.

“We’ve received 86 calls from patients in the past 10 minutes,” he says. “There are three calls waiting and we have answered 83 calls. The average delay before answering has been 2.4 seconds. We’ve answered 2,267 calls since midnight. Our average answering time has been eight seconds.”

Next door, Sheena McLeod, a senior nurse from Australia, with a background in HIV medicine and a bedside manner that would shame Florence Nightingale, is picking up calls. The wife of a man in his sixties from Dundee is on the phone. She tells McLeod that her husband, who has been treated for cancer of the oesophagus, is incoherent, hasn’t eaten for a week, is behaving irrationally and has taken off all his clothes because he is so hot. He does not want to see a doctor, however, and his wife is not keen to let him know she has called NHS 24. McLeod decides that the patient needs a home visit within two hours. She puts the patient’s wife on hold while she makes the arrangements.

“I’m really glad you got in touch with us,” she tells the caller. “It was the right thing to do. If he gets any worse in the meantime, or even after the doctor has been, call us straight back.”

I am beginning to wonder why anybody bothers battling with their GP’s receptionist during the week. Here, sympathetic senior nurses are arranging out-of-hours appointments and home visits with great efficiency. As if reading my mind, one of the nurses reports that her last caller — ringing from one of the most prestigious addresses in the country — said she was “much too busy” to take her sick child to the GP during the week. She, too, gets an out-of-hours appointment for that afternoon. It sounds like an abuse of the system.

Needless to say, this level of service does not come cheap. A joint independent report published in December into NHS 24 discovered that the cost of a call to the service was a staggering £35.69. The total cost to the taxpayer of NHS 24, so far, is more than £126m; the equivalent of almost £9 for every man, woman and child in Scotland.

Given that the cost of a 10-minute appointment with your GP is about £22, given that the NHS in Scotland last year had a £32m deficit, and given that 60% of all calls to NHS 24 result in a face-to-face consultation with a doctor, it is hard not to question the economics, especially when at peak holiday times GPs have been drafted in to man the phones on salaries of more than £1,000 a shift. So far there is no indication the demand on accident and emergency or the ambulance service is falling as a result of NHS 24, leading some observers to see it as an expensive extra tier of service used mainly by the worried well.

“It does sound awfully expensive,” says Dr Andrew Walker, a health economist with Glasgow University’s department of biostatistics. “You do have to question whether it is the best use of such highly qualified nurses.”

Times Recommends

• Donor organ allowed child’s own heart to heal
• Michael Jackson: Not the final chapter
• Brüno toned down for 15 rating