Thirty years since it first aired, voice of Mrs T is still unsure whether satirical puppet show helped or harmed her reputation

She used the men's toilets, smoked a cigar and ate her steak raw – the vegetables could look after themselves. Thirty years on, the voice of Margaret Thatcher believes the jury is still out on whether Spitting Image helped or harmed her reputation.

"One of the big questions which is still unanswered is whether we did her a favour or not in the sense of portraying her as strong and dominant," said Steve Nallon.

He was with Roger Law on Tuesday at the opening of a new exhibition at the Cartoon Museum in London telling the story of one of the most original and funniest TV satire shows ever made.

It was on 26 February 1984 that the caricatures of Peter Fluck and Law were brought to life in puppet form for Spitting Image. The show ran on ITV for 18 series, bringing to the world a jive-talking banjo-strumming Pope John Paul II, a CND supporting Queen, a leather-clad Norman Tebbit and many more.

The exhibition includes puppets, and perhaps offers the only ever opportunity to see Osama Bin Laden, the Queen Mother, Alan Bennett, Jacques Chirac, Will Carling, Roy Hattersley and Princess Diana together.

Thatcher gets her own display cabinet. Nallon said it took a while to get the voice for her right, adding that it became more aggressive as the programmes went on. One crucial decision half way through the first series was dressing her in a suit – and after that she would use the gents.

From then on the whole country assumed that the programme was factually correct in portraying Thatcher as a power mad bully who dominated her craven, terrified cabinet.

Nallon said: "Everybody remembers the sketch of Mrs Thatcher peeing next to Heseltine and Heseltine saying, 'I can never go when she's standing next to me.'"

Law said the programme probably did not change anything substantial except perhaps that children of the 1980s knew who ran the country. "I've got eight grandchildren and not one of them knows who's in the Labour or Tory party," he said.

The show includes many original drawings and scenes created for magazines such as Prince Andrew as a naked hunk with 2lb of Cumberland sausage between his legs. "The shoot went on half a day," said Law "We were so hungry we ate it."

Then there are letters of complaint including a particularly forthright one written in October 1986 and diligently filed away. "You ignorant bastards are just pisstakers out of our country and a pack of reds. We are going to march in protest," it read.

In truth, said Law, not that many people were upset. "The English care about literature and gardening. They don't give a shit about what goes on the television most of the time."

Ironically the show became everything Thatcher could hope for – it was a global British export with spin-offs in countries including Hungary, Germany, Italy, Portugal Japan, Russia and Greece where one man did all the voices. Versions of it continue in Iran, Kenya and South Africa where Law was last year only to hear the Spitting Image song I've Never Met a Nice South African being played in the production company offices.

There have been attempts to bring back Spitting Image on TV and the web, but Law said it was too expensive – it had high production values and they were lucky to have the support and financial backing of Central TV, he said.

Most of the puppets have dispersed around the world after two online auctions in 2000 and 2001, and the foam used in their construction can easily degrade. The cracks in the Bennett puppet were plain to see at the Cartoon Museum but as Law pointed out: "Have you seen Alan Bennett recently? Most of the puppets look better than the people are now."

• Spitting Image – from Start to Finish is at the Cartoon Museum, London from 26 February to 8 June.

• This article was amended on 26 February 2014 to correct spelling in paragraphs 6 and 9.