A few years ago, there was serious talk of Sandi Toksvig standing for election as a Liberal Democrat MP. She would have been brilliant, making light work of constituents' plumbing problems, with the sort of brisk efficiency that seems particular to middle-aged mothers. Anyone who spat out their tea – in horror or delight – on hearing her say "the Tories put the n into cuts" on Radio 4's News Quiz could imagine her shining at prime minister's questions.

But Toksvig has washed her hands of the party. "Do you remember the Lib Dems?" she says. "They were great, weren't they? I don't support anybody now." Are they still counting on her support? A raised eyebrow. "I don't know if they're counting on anybody. It will be interesting to see, when they go to the polls, what has happened. I'm political in the sense that there's much to be done, but I'm apolitical in the sense that I don't think there's a party that represents anything I believe in."

We are talking in her publishers' boardroom: Toksvig's new novel Valentine Grey, about a young woman who goes to fight in the Boer war in a bicycling regiment, is published next week. She has been writing the book on and off for four years, interested in the Boer war "because it was the first war where the average soldier was literate and they wrote home and said, 'This is rubbish, a really bad war and conditions are terrible and we shouldn't be here.' It did mark the beginning of the end of empire. It was the war where the public started to say, 'What are they doing there? What are they dying for?'" In person, Toksvig's voice is quieter than on TV or radio, with a gentle Scandi lilt. (She is also more Danish than the clipped broadcast tones would suggest.) She is warm, funny, interested in everything; I get the impression you could put a brick in front of her and she would become fascinated .

Her character, Valentine, is a young woman who grew up in India but is sent to England; her escape to war is also an escape from the restrictive ideas of her time. "I started, just by chance, reading about various women who had served as soldiers – Hannah Snell in particular, who became a marine, and served for six years on a boat called the Swallow and none of the men suspected. I became interested in the idea that a woman might desire a bicycle so much. In 1899, the bicycle was such a significant invention for women, in terms of changing their clothing and the literal liberation from the house."

Did it make her think about how far feminism has come? "I still think we have to pay attention," she says. "It's almost as if women are allowed to get to a certain stage and then something happens. I worry about role models. Although I'm sure she's completely charming and delightful, I'm not sure if Kate Middleton might be the best role model. This is a person who has got where she is by marriage, a person whose weight, clothing, hair we worry about – we don't worry about what she's thinking. I'd much rather see a feisty, intellectual powerhouse on the front of magazines. I would love to see a 23-year-old Germaine Greer or Maya Angelou, who is there because they have something to say or have done something amazing."

Famously, Toksvig filmed one of the first pilots for Have I Got News for You, before it was decided that the show should be fronted by a man. Even when she became the presenter of the News Quiz, there were rumblings that the audience weren't ready. "They had thought it would bring shock to the shires, but apparently it's OK." The News Quiz is often held up as the panel show that attracts more female comedians than any other. "When people say, 'There aren't enough women on panel shows', the answer is to make the host a woman," Toksvig says. "She will automatically reduce the amount of competition in the air, and if the boys get out of hand I quickly tell them. I always make sure, if we have a new person – man or woman – there is space for them to say what they want. On many of the panel shows, there isn't a space. After a while, if you are a woman, and you're having to compete for room you just stop bothering and think 'I can't be arsed'."

We talk about television critic AA Gill's recent attack on classicist Mary Beard's appearance – "Isn't it a sign of the times? How dare they. What about the intellectual content?" – and the fact that Toksvig remains one of the few middle-aged women on television. "We can't just have very good-looking women who at the age of 30 appear to go into some twilight home. Society is made up of far more interesting individuals. I think the public is happy to have a whole range of people."

Born in Denmark, Toksvig grew up in the US, where her father, a foreign correspondent, had moved for work. She was sent to an English boarding school at 14 after being thrown out of a number of US schools ("the last one, they had a ridiculous rule about being there every day!"). Boarding school was "a compete culture shock. I had a very thick New York accent, and the girls were horrid." She wasn't bullied, she was ignored. "And I had come from a very freewheeling society and suddenly there was no TV, radio, music, art. You were allowed three things on your dressing table and one had to be a hairbrush and one had to be a framed picture of your parents. Really, a kind of life that is hard now to imagine you would put children through. Fortunately, I like to read and I retreated into my books."

At Cambridge, although she was a member of Footlights, her aim was to be a human rights lawyer. (She thinks it still might be, describing her career as "one long gap year".) In her final year, she became very ill, and her then girlfriend came to stay. When her college discovered this, they threatened to throw Toksvig out. "Lots of girls had boys living [there] full-time. They took three weeks to decide whether or not I could stay and they finally said I could 'in light of my excellent academic record'. So it's OK to be gay, as long as you're clever. I still, to this day, find it hard to forgive them. It was so naked in its prejudice. It hurt me terribly because I was passionate about the college [Girton]."

Is she bored of being asked about being gay? She doesn't show any sign of it. There are so few gay women with a public profile, I say: why does she think this is? "Trust me, there ought to be more. I think it's still not quite the thing. I do know one or two people who are afraid to come out. I've never been so ambitious that I thought, 'What will I do if this damages my career?' When I came out, people said your career is over, and I said,'I'm fine with that. I have three children and I brought them into the world and I have a responsibility to make sure they can live with their heads held up high, so I'll do something else.'"

In the early 90s, Toksvig was told she was about to be outed by a tabloid newspaper. She and her then partner, and their children, went into hiding. "We had death threats, mainly from very religious people who very kindly were going to kill me on God's behalf because he's busy." She laughs, then adds: "It was scary. I'm on kissing terms with the police hate crime squad." She brightens. "I'll tell you what makes it worth it. I still get young women who come up to me and say, 'Thank you, because I was able to say to my mum, ah, but you like Sandi and her life is OK.' I might not have chosen [to be gay] – it's hard to be outside of the norm – but I'm glad because it has made me more tolerant. Now I'm 54, I'm delighted and happy – but the years when I was scared for my children, I could have done without that."

Does she think gay people in the public eye have a responsibility to come out? "I think the only time you have a duty to come out is if you're being hypocritical – if you're an MP and voting against gay people's rights. I don't think it's a duty, but I think it's a shame for people who could be wonderful role models to take the plaudits and not know that it comes with responsibility."

Toksvig wants to write a followup novel about the suffragette movement, and another children's book. She is writing a pilot with the comedian Susan Calman. "I've got a musical and a play I'm commissioned to do. I'm writing a book about modern manners. I'm finishing a book that comes out in October, which I think will be called Heroines and Harridans. I've worked with a friend who is an illustrator and it's great women in history, randomly selected by me." She is touring a one-woman show based on Valentine Grey, doing a Channel 4 quiz show, and working on the West End transfer of her play Bully Boy (about the lack of mental health care for returning soldiers); then there is the return of the News Quiz. She produces new drama for Sky Arts, and in October will take up the post of chancellor at Portsmouth University. She makes Stephen Fry look like a layabout.

Why does she work so much? She laughs and says her partner Debbie, a psychotherapist, asked her that this morning. "It's partly that the world is so interesting. I do have a tremendous sense of there not being enough time to meet all the people I want to meet, travel, read all the books, listen to all the music. I genuinely do not understand boredom. Debbie sometimes says to me, 'Darling, could you not have your first idea before I've had a cup of tea?'"

I read an interview with Toksvig's sister, a librettist, who said she thought Sandi "is basically a happy person, but I don't think she is fulfilled". She laughs. "That's probably true. I think there's something we all have to do and I haven't done that yet. I'm still looking." She looks slightly troubled before saying brightly, "But I don't know what you do when you find it and you do it. Maybe just have a sit down."

• Valentine Grey £16.99 (Virago) is published on September 6. Her show Sandi Toksvig Live is now touring. Details: sanditoksvig.com