THERE is a moment when it seems Susan Calman's emotions could go one of two ways, hovering precariously between frustration and tears.

A slight wobble creeps into her voice, but she deftly bats it away.

We are talking about what it's like to be a gay woman in 2012. The comedian and daughter of Sir Kenneth Calman is referring to a story that has been in the news on the morning we meet, with Cardinal Keith O'Brien, leader of Scotland's Catholics, having called for a referendum on gay marriage.

Calman, who got "civil partnered" to her girlfriend of nine years in June, says she can't believe it is still an issue in the present day. She looks at me, a mixture of sadness, anger and weary resignation in her voice. "I would never dispute anyone's right to an opinion, absolutely not," she says. "But I feel like my side of it hasn't been given as much air space because we haven't got such an interesting story. I simply want equality. I want the same as what my sister or brother have."

It is a topic Calman will address in her forthcoming Edinburgh Fringe show, This Lady's Not For Turning Either, her most potent – and potentially inflammatory – yet. It's also her bravest, an act of defiance against the almost daily headlines that Calman views as an attack not only on her sexuality, but her rights as a woman and human being. The show will cut to the crux of the debate over gay marriage currently raging in Scotland, laying bare her own life in the process.

"I learned my politics through comedy in the eighties," she says. "I wanted to do a show where, first of all people learn something about me – whether they like it or not – but with a bit more politics than I normally do. I want to tell everyone what it was like growing up in Glasgow and coming out. Not easy – and still not necessarily easy."

She breaks off as our tea arrives. As the waitress retreats, Calman pours herself a cup, settles back into her seat and picks up where she left off. "You can go and see a male comedian talk about getting married – it's the same thing," she continues. "I want people to listen to that viewpoint without any hatred or negativity. I think we should all be able to live together. I'm old- fashioned that way, possibly a hippy.

"Not all gay people agree with marriage and, again, that's fine. This is a personal viewpoint. But I think in 2012 we shouldn't be having these discussions.'"

A little over a week after our meeting, Calman's voice drifts down the line. It's the day of the Scottish Government's announcement that they will legislate for equal marriage. "I'm cautiously pleased," she says. "It's undoubtedly a cause for celebration, but I believe there is a long way to go. I hope I'm forgiven for being a little reticent. I'm delighted but will keep fighting until it is actually passed as legislation."

The youngest of three children, Calman, 37, grew up in Glasgow. Her father, Sir Kenneth, is the former chief medical officer of Scotland and chairman of the Commission On Scottish Devolution, her mother, Lady Anne, a retired teacher who is now chair of NHS County Durham.

Their daughter was a late bloomer when it came to comedy. It wasn't until the age of 30 that she packed in her job as a corporate lawyer for the stand-up circuit.

She cuts a diminutive figure, barely 4ft 11in in her stocking soles, with a warmth and endearing candour that she professes to be "a terrible fault". Calman has a habit of frowning alarmingly as I pose each question, her forehead ruching like a freshly ploughed field. Hers is a wry turn of phrase: she says growing up as a gay woman in Glasgow was about as easy as being "a vegan abattoir worker".

This was a period when Section 28 (2A in Scotland), the legislation banning the promotion of homosexuality, was in full force. "It was a horrible time," she says. "No-one should be growing up with any bigotry attached to them for any reason: colour, religion or sexuality. Someone said that gay marriage would be a shame on our country – and that is a very hurtful thing." There is a catch in her voice. "It is good to have the debate. I get very angry about it – but I don't think anger helps anyone so you do have to stay calm."

Even so, there is a fire in her belly as she talks about attempts by the Catholic Church in Scotland to push for a referendum on same-sex marriage. "I don't believe we can change certain people's opinions," she says. "I don't think that, even if I sat in a room with Cardinal O'Brien, I could change his mind. I feel very sad. You wouldn't have a referendum on someone of a different colour. It wouldn't even be acceptable to suggest to have a referendum on that, but it's acceptable to say this.

"It is maintaining ignorance. It's the same stuff I heard growing up about the 'Aids plague' and the fear people had then about gay people."

Calman describes her own teenage years as "quite isolating" and "like a true-life movie on Channel 5". She sought solace in the comedy of Joyce Grenfell, Hattie Jacques, French and Saunders, and Victoria Wood. Yet somehow Calman cheerily manages to extract a positive. "Comedians tend to be outsiders," she says, smiling. "Most of us have a sense of isolation and being on the outside looking in. So it was a good thing for my comedy, not being the most popular girl in school."

Calman welcomes what she views as a chance to "hold a mirror up" to the stigma that many gay people still face, not least talking candidly about the ceremony for her own civil partnership to Lee, a lawyer.

"We weren't allowed Pam Ayres's I'll Marry You because it's against the law to have that poem – it's not just about religion, it's about the word 'marry,'" she says. "Sometimes if you tell people: 'This is what our ceremony had to be,' they'll go: 'But that's silly' – and yes, it is silly. Having been through it I can point those things out.

"I will tell stories about my relationship and people say: 'Oh, but that's what I do with my husband, boyfriend, wife or girlfriend.' I think sometimes people perhaps expect us to be living in a very different way. But it's nine years, three cats and the same problems as everyone else."

Calman smiles blissfully when asked about Lee, 40, whom she met on a blind date. "I very much fell in love with her the first moment I saw her – it took her slightly longer," she says. "I thought she had the kindest face I had ever seen. So we met, got together – and she changed my life because she encouraged me to give up my job."

It was a leap of faith that saw Calman make the transition from legal brain to comedy hopeful. In her first year of plugging away on the stand-up circuit she made about £250. But Calman was determined. "All the people who said it was a stupid thing to do, I didn't want them to be right," she says.

Seven years on, her tenacity has paid off. Calman is a regular on Radio 4's The News Quiz, has guest-presented Woman's Hour, appeared on comedy panel shows QI and Have I Got News For You, and is a frequent stand-in for Fred MacAulay on BBC Radio Scotland. She is carving out a career in acting and writing sitcoms too, not least working alongside Jennifer Saunders and Sandi Toksvig.

Calman appears caught between forced, airy nonchalance and wanting to jump up and down in excitement. "It's getting there, no question," she says, cautiously. "You have got to keep working – if you become complacent, that's when it goes wrong. Sometimes I do need to stop and take stock because I work all the time. I'm a workaholic – nothing is ever good enough."

Workaholic is a word that could have been coined for her. She admits to keeping Thatcher-esque hours, averaging only four hours' sleep most nights.

Calman has forged a strong working relationship with her fellow female comedians, describing Toksvig as "a delight" and Saunders as "lovely", while Jo Brand is an "absolute doll" and Sue Perkins "so supportive".

She gets frustrated by attempts from some quarters to pit women comedians against each other. Ditto the oft trotted out notion that women simply aren't as funny as men. "People say 'women comedians' like we are all the same – and we are not," she says. "I don't like some male comics, but that doesn't make me think I don't like all male comics. Tina Fey said it brilliantly: she doesn't like Chinese food but doesn't write endless articles about how she doesn't like Chinese food.

"I said in my Edinburgh Festival brochure entry: 'Don't come if you think women aren't funny.' I'm a comedian first and foremost and if you don't like what I do, it may not just be because I'm a woman."

What about hecklers – do they ever target Calman for being a woman or for her sexuality? "I have only had two or three instances where the heckling has been really personal," she says. "One was at a gig in Newcastle where a guy in the front row said: 'I'm going to rape you.'" Calman nods as my jaw hits the floor. "Which was an awkward situation. He wasn't drunk. He thought it was funny."

How does she handle that? Is it a case of the show must go on? "No – one never continues with the show at that point," she interjects. "What you do is have a discussion with the guy about why he thinks that is an acceptable thing to say. There are heckles you ignore, but that isn't one of them. Instead you say: 'Do you have a sister, mother or girlfriend? Would you like that to be shouted at them? Do you think it's appropriate?'" She takes a sip of tea. "I'm not saying it's the funniest set I ever did, but you can't ignore something like that."

The way Calman sees it, it's a case of "my house, my rules" – and that includes banning her parents from her shows. "I would get so nervous if they were there," she says. "They – and their approval – mean such a lot to me. I couldn't stand it if they were there and I had a bad show. They keep saying, 'We'll sneak in.' But if I saw them in the audience it would be like: 'Right, no, c'mon, out you go -'"

Given the nature of her father's high-profile career – his many hats include chancellor of Glasgow University – was it hard growing up with a parent in the public eye? "I used to meet people and they would say, 'Oh, are you Kenneth Calman's daughter?'" she says. "It was one of my happiest moments when he texted me recently to say someone had asked him: 'Are you Susan Calman's dad?'"

I have that awful moment all interviewers dread when my mind goes blank, her father's most recent appointment falling clean out of my head. I can't even fudge it. Fortunately Calman helps me out. "I think he's the head of the National Trust For Scotland now?" she deadpans, trying not to laugh and waving away my embarrassment. "He does various things – he always has something on the go."

She certainly seems to have inherited her father's incredible work ethic. "I'm hugely like him – and becoming more like him," she confirms. "We are both always working. I have a notebook like him and write things down, my list of what I need to do."

Their strong father-daughter bond is palpable as Calman speaks. When she got "civil partnered" to Lee, he proudly walked her down the aisle. "He did a speech and got quite emotional. It was lovely," she says.

The couple's bash at City Halls and the Old Fruitmarket in Glasgow was "pretty full-on," she admits. "We wore suits – although not matching," she hastily adds. "We went for a traditional grey with some nice shirts and comfortable shoes – I have bad feet so I couldn't have worn heels."

Their first dance was to Bring Me Sunshine by Morecambe and Wise. "The lyrics are beautiful," she says. "We did a wee dance and then we left the dancefloor doing a Morecambe and Wise-style skip."

Calman is yet to whisk away her "wife" – as she affectionately refers to Lee – off on honeymoon. "I have so much on at the moment," she says. "I'm not good at taking holidays." I don't envisage her being able to sit still long enough for a typical beach honeymoon, I say. "No," she cheerily confirms. "I don't fly – this is one of the problems. I think we are going to go on Eurostar somewhere. I have been away once in nine years on a plane to Cyprus."

How did that pan out? "Not well. I spent five hours crying on the way there and five hours crying on the way back." That wouldn't make an ideal start to a honeymoon? "Er, no – and that was with diazepam. I just don't like flying."

While she recently become proud aunt to her sister's daughter, Calman has no plans for children of her own. "Lee and I have discussed it and, no, I don't think so, we are perfectly happy," she says. "We have godchildren and nieces – and will be fantastic godmothers and aunties. If we had children, one of us would have to give up our job. Neither of us would want to deny the other that opportunity."

She has her hands full with feline trio Oscar, Pickle and Muppet. "I know I talk about my cats a lot, but I love them dearly," she laughs. When Calman is holed up writing in isolation they are her sole companions for much of the day. "It can get a bit lonely," she says. "The cats and I have lunch together, a wee chat about what's on the news. They are very much my company."

Her star may be firmly in the ascent, but don't expect Calman to be playing any stadium tours soon. "I like the fact people do these gigs and it's very rock 'n' roll, but I prefer venues of 120-150 people," she says. "I like to see people and being able to talk to every one of them if I wanted. So I will probably be very poor – but if I wanted money I would have stayed a lawyer." 

Susan Calman: This Lady's Not For Turning Either is at Underbelly, Bristo Square, as part of the Edinburgh Festival Fringe, from August 1-27. For tickets, visit www.edfringe.com or call 0131 226 0000.