It was Cath Tate’s image of a Tory prime minister pickpocketing a young mum that first showed her people would buy the cards she designed. In the early 80s, Tate took a photo of her friend and toddler, cut it into a montage with Thatcher pinching the young mother’s purse and sold the result as a greetings card with the slogan: “Prevent street crime”.

“It was the first one that really took off,” says Tate, who is now in her 60s. She started out selling postcards with anti-Thatcher, pro-feminist messages, but never imagined she would end up with her birthday cards in gift shops all over the country. Producing the cards was a part-time job while she raised her children but she gradually found that it was getting easier and easier to pay the bills. “It was quite a financial struggle when the children were little,” she says. “It didn’t happen overnight.”

After the Thatcher image sold so well in alternative bookshops, she was introduced to a group of female cartoonists and began producing their work as cards. “I realised – the more I looked into it – how good some of the women cartoonists were,” she says. She had the idea of an exhibition of female cartoonists’ work but, at the time, no one was interested. Now, about 30 years later, it has finally happened, at the Cartoon Museum in London.

The exhibition, The Inking Woman, goes back to Mary Darly’s satirical prints in the 1700s, includes feminist cartoons of the 1980s and 90s, and continues up to the present day. The most familiar one from the 1980s is probably Viv Quillin’s drawing of a smug, Garfield-esque moggy with the thought bubble: “Behind every gifted woman, there’s often a rather talented cat”, which Tate produced as a greetings card.

After the female cartoonist cards, the next big hit for Cath Tate Cards, by then a professional enterprise, was combining images from 70s knitting patterns with wry slogans. And then her enduring – and much imitated – old family photograph range. It was this that sent the business into the big league. In the late 90s, John Lewis and Paperchase started to take her cards and, by the early 00s, the company’s annual turnover hit £1m.

Around that time she took her teenage children on holiday to the US. “We went up the Empire State Building and I remember thinking: ‘God, I’ve actually been able to afford to bring them over here on the strength of the business.’ It was such a feeling of achievement.”

Tate credits her success with knowing the market: her friends are into yoga and therapy, and like making fun of themselves and each other. “Not everybody wants to send silly flowers on cards,” she says. “I want to send cards to friends that are a bit rude because it’s a way of showing affection. If someone sends me a card that’s too schmaltzy, I get a bit suspicious.”

She was a secondary school teacher before she had children and, as a child of the 60s and a radical leftwinger, never expected to become a businesswoman. She credits her politics as having driven her success. “I went to university at a time when there was a lot of student unrest and I got politicised then. That has coloured what I’ve done since.”

Has she developed more sympathy for Tory policies as she has grown a successful business? “No, because on the whole Tory policies help the rich; they don’t help businesses. They don’t see the bigger picture. They’re actually not particularly conducive to business because, as an employer, I’m dependent on being able to employ well-educated youngsters. If money has not been put into education, then employees suffer. As an employer and a business, you’re dependent on your employees having decent healthcare and decent housing.”

Cath Tate Cards’ newest range features historical portraits; for example, a painting of George Washington saying: “For your birthday, we have a fruitcake for you in the White House.” She got the idea for them after hearing the news that the remains of Richard III were discovered under a car park in Leicester. “I’ve got a friend who lives in Leicester, and just the idea of Richard III being found in a council car park was so funny.” The caption for the Richard III card is: “500 years and still nowhere to park …”

There have been times when she has thought about giving up, usually when a popular range has started to drop, but then another idea has come along. “I’ve been quite lucky really,” she says modestly, but she recognises that producing cards that people want to buy is a highly tuned skill. “Somehow one has to produce cards that hit the moment,” she says. “It’s quite a subtle thing, what people think of as funny. You have to be plugged in to what is really getting at people.”

•The Inking Woman is at the Cartoon Museum, London, until 23 July.