At the beginning of the year, I discovered a programme my daughter and I can watch together. She has just gone 12. This may not sound like a big deal, but trust me, it is. There are only so many old episodes of Friends I can watch. She loves New Girl, whereas I find that Zooey woman the most irritating person in the world. Life is too short to try to appear ironic and hip and endure the moneyed banality of the nightmarish Kardashians. These days, apparently, I even ruin The X Factor for her by repeatedly asking, "What happened to … ?" The Big Bang Theory is about as good as its gets.

So when we started watching a drama series about midwives in the 1950s, I was suprised she stuck with it. And I was surprised at how much I was moved by it. Clearly we were not alone: Call the Midwife is BBC1's most popular drama in more than a decade, the final episode of its first series pulling in more than 9 million viewers. A mark of its success is that the Christmas special will be the centrepiece of the schedule.

Though popular, it is continually referred to in all the TV previews as the "grittier" choice. I wonder what this is code for; certainly my daughter finds most of the graphic childbirth scenes "completely revolting" while I constantly get something in my eye while it's on. She also hates "all nuns", and yet we watched every episode, so something clever is going on. On paper, midwives, poverty and childbirth do not sound like primetime viewing. Yet this slice of social history ends up being a great watch.

So when I was offered the chance to see it being made, I was intrigued – and to be honest, my daughter was desperate to meet Miranda Hart, one of its stars. Off we went on a very cold day to the set, which is in fact a former convent that has been used as a hospital. It certainly had the gloom of an institution, and managed that trick of being colder inside than outside. As we tiptoed over to watch the filming, Jenny Agutter, who plays Sister Julienne, was waiting too. Dressed in a nun's habit with an anorak over the top, she was clearly ill. In between takes she had to keep going inside for fits of coughing, but on set she still managed to project the aura of kindness and stillness that makes her so right for the part. "I guess," she ventured, looking out at the 1950s set, "this WAS the big society, where there was caring, not by just a few, but by many."

Call the Midwife is based on the memoirs of Jennifer Worth and – scripted by Heidi Thomas, who wrote Cranford – it tells the true story of a young midwife going into Poplar in the East End of London in the 1950s. It's a tough landscape of slums and bombsites used as adventure playgrounds; there are knifings, brothels and real poverty. It's a world where there is much talk among women of "safe periods, slippery elm gin and ginger, of water douches," according to Worth, who adds, " ... from what I heard most men absolutely refuse to wear a sheath". At 22, Jenny Lee goes to work with the midwives of St Raymond Nonnatus, an order of nuns who had worked in London's slums for years, through outbreaks of cholera and typhoid, and delivered babies in air-raid shelters and crypts during the Blitz. These nuns were qualified nurses, and respected and admired by all.

I see an actor in a huddle with Terri Coates, the midwifery consultant on the series. "Women in labour make a sound from much lower in their bodies," she tells me. "I have to explain sometimes. It's very guttural. It can't come from the top of their vocal range". There is some practice of animal-sounding groans.

Coates is more than a consultant – it was her article in the RCM Midwives Journal in 1998, about the absence of midwives in literature, from Dickens to Chekhov to Dostoevsky, that inspired Worth to write her memoirs. Why were midwives portrayed as "shadowy figures", when in fact they see it all? It is this that is at the centre of the drama.

Coates soon starts making my daughter feel sick when she takes out a doll of a newborn, which we christen Scary Baby. The birth scenes are realistic, with lots of coaching from Coates. Scary Baby is covered in oil and fake blood, and is produced by Coates while squatting under an actor's legs, with a real baby substituted at the key point. "It's disgusting," says my daughter. "It's childbirth," I say, though I too am fairly disturbed by Scary Baby. All day long, I cannot tell which actors are actually pregnant and which have pregnant suits on. Coates is on top of every detail, at one point showing the cast how to do a proper abdominal examination. This level of authenticity clearly works, and while we are now used to seeing real childbirth on TV in progammes such as One Born Every Minute, what is so powerful about Call the Midwife is precisely what it shows about labour and birth long before epidurals: women suffer. There is no sugar-coating here.

It is, as Thomas tells me, absolutely about the "reality of childbirth". The line that got to her most, and made her want to adapt Worth's memoirs, was the one about just how much women have to bear. More than that, she responded to a story written "through the prism of kinship, deeply felt". There is a longing for "community" that Call the Midwife represents, and yet whenever this nostalgia gets mawkish, we are brought up short with a stab of recognition about the harshness of that time.

The skill of Thomas's writing is apparent, as there isn't a huge narrative thrust. Worth's story is lucid but fragmentary, and producer Pippa Harris's belief that Thomas should take it on has proved spot-on. That this show is not only a huge hit here, but now also in the US, underlines this. Some have complained that it all looks too clean, and that the real East End would have looked much dirtier. In fact, it is all lovingly reconstructed from photographs of the time.

What it does show, though, via a look at another era – a time before washing machines and the pill, those great liberators of women – is how far we have come in so many ways. It works as a short, sharp shock to anyone who would take us back to these times. Prostitution, abortion and forced adoption, all stirred up with a bit of syphilitic discharge, undercuts any Hovis-advert depiction of postwar working-class life. Such hardship is made manageable as we encounter this deprivation through the eyes of Jenny Lee, voiced in the series by Vanessa Redgrave. Lee, who is shocked at what she sees, learns that the job of the midwife is to "make a woman feel safe". Social realism is lightened by a certain silliness and, of course, by the award-winning performance from Hart as Chummy. We watch her do take after take of a scene out in the cold, but the warmth between the cast feels real. Hart whoops with delight on seeing Judy Parfitt, who plays Sister Monica Joan. "We just make each other honk with laughter."

Outside, a scene is being filmed for the second series, which will be shown next year. A woman is in full labour, and the nuns are helping her struggle up the stairs while her friend draws heavily on a cigarette. I chat to Pam Ferris (Sister Evangelina), who knows what a great role this is. She was bored with playing "what my husband calls super-mum" parts. She thought twice about playing a Catholic nun, but she could see these nuns were "such interesting women, who functioned as practising medics, who had autonomy". Ferris sums it all up as "blood, guts and faith".

By now, we are huddled around a heater. Any idea that filming may be glamorous has been erased from my daughter's head. While I chat to the producer about audience figures and US acclaim – not one bad review – my daughter looks spectacularly bored. "When am I going to talk to Miranda?"

The first series ended with Hart's character, Chummy Brown, marrying PC Noakes, and one standout scene was when she overcame her self-doubt to deliver a breech birth. "Bravo, Betty, Bravo," she gently encouraged as she rotated the baby's head, in a moment that captured the agony and ecstasy of birth. It was Worth herself who first suggested Hart for the part of Chummy. She had no idea Hart was a star. Worth – who has since died of cancer – called Harris to tell her she had spotted someone. "I doubt you'll have heard of her, but her name is Miranda Hart. She's six foot two and falls over splendidly. I think she'd be quite marvellous."

Indeed she is, and when we do grab a moment with her, in between her instructions to her co-stars ("Come on Agutter") it's clear she has always thought of herself as a serious actor. Why do we like to watch such difficult stuff, I ask her. "Perhaps you are all mental?" she says of the Call the Midwife audience. Like every other member of this ensemble, however, she seems unsuprised at the show's success. My daughter does the fannish thing of not actually speaking to Hart, and instead spends the whole time texting "I am with Miranda" to her mates. Hart seems to get this, and poses for a picture with her. Then she's called for another lesson in obstetrics in a dingy reconstructed 1950s dining room.

Later we watch the Christmas special, and there is something rather meditative about it, especially when the nuns sing together. The whole pace of the show is slow compared to much drama today. But it works.

A sure sign that it does is the fact that it worries the Daily Mail. A "TV insider" has told them it will be shown pre-watershed, even though it features "two harrowing childbirth scenes". This anxious "insider" goes on to say that the programme may have "a huge and loyal female audience, but do two agonised births go with turkey sandwiches, boxes of chocolates, champagne and men wielding the remote control?" What were they thinking, talking about childbirth on Christmas day?

Actually, it's not the birth scenes that are harrowing but the echoes of the workhouse. Like the Olympic opening ceremony did, Call the Midwife openly celebrates the birth of the NHS. Suddenly people could have walking sticks and glasses, and the poorest could see a doctor. This was some kind of miracle.

Yes, there is sentimentality, but the series continues to uncover social history, the hidden history of so many women's lives. That such a popular drama does this so fearlessly is rather magnificent. And beautiful: moving between earthiness and otherworldliness, the Christmas episode is special. And the only spoiler I will give you is that, among the sadness, Pam Ferris does some brilliant farting.

Perhaps this and the Scary Baby scenes are why all the critics keep talking about grit. I don't think that's the right word for this show at all. Grit is easy. What the show has that makes it compelling is something much harder to define: it has real soul.

Call the Midwife is on BBC1 on Christmas Day at 7.30pm