Today he and his Channel 4 show offer what is probably the most biting satire on British television or radio; and satire not about other television programmes (Steve Coogan, Harry Enfield) but about politics; and not about the comic-book personality of politicians (Spitting Image) but about the deeds they do. Those old enough are harking back to TW3, the high-water mark of British satire, and the part it played in the downfall of Harold Macmillan.
That is high praise, but perhaps very little of it is earned by Bremner's undoubted talents as an impersonator. His scripts and sketches are not just broadly political, but intensely, topically so. They deal in scathing detail with quangos, with an economy that is recovering only because the Government abandoned its economic policy, with the internal market in health, and sleaze.
Last week, Bremner spent three-quarters of an hour with Mohamed Al Fayed. He also met Marjorie Wallace, the chief executive of the mental health charity Sane, known for her forceful views on care in the community. And in the weeks preceding the series, he and his team lunched lobby correspondents and economics editors, and cultivated moles in the Labour and Conservative parties.
Bremner is producing investigative satire of a kind that is not seen elsewhere on television. But the best satire is often as shocking as it is funny. When Jonathan Swift proposed eating babies as a cure for famine, it is doubtful whether people fell about. And to point out that John MacGregor, who used to be Secretary of State for Transport, is now on the board of Hill Samuel, one of the Government's key advisers on rail privatisation, is not necessarily side-splitting.
Bremner's technical excellence can have drawbacks: it is easy to be enchanted and distracted from the satire by the sheer scale of his accomplishment. And where fictional characters (such as the monstrous personalities who front The Day Today) can do extreme and terrible things to make a comic point, he has to stay fairly close to his originals. So the funniest bits - like the spoof documentary on the seedy underworld of Labour's rent-an-idea boys - sometimes don't rely on impressions at all.
But throughout, the satire is ferociously well-informed (sometimes anticipating press excitement. Last year, he was going for quangos; this year everyone is). His main interest seems to be in truth-telling: 'Satire is often confused with irreverence,' he says. 'Spitting Image feels rather like graffiti: perfunctory and unfocused, because it's poorly researched.
Satire isn't about John Major pushing peas round a plate.' What it is about, he thinks, is 'the information gap - between what people know and what's really happening. And the more you look at health or transport policy, the more you want to ask: does it have to be like this?'
HE ARRIVES at speed, bulky but cherubic, bounding up the stairs to the offices of his production company in a scruffy building in Soho. He owns the company with his agent and two producers; they set it up to make the current series. Too many second-hand desks are crammed into not quite enough space, crates of Jacob's Creek and Beck's are piled in the tiny kitchen.
Bremner talks hectically, in clipped, public school tones. He has wedged this interview between a costume fitting and trying to decide on this week's lead stories. 'Al Fayed, the Commission on Social Justice, pensions, or the Serious Fraud Office?' he speculates with an anxious air, shuffling papers.
He was born in Edinburgh in 1961. His father was a Scottish retired army officer (his mother is English) who worked for the Cancer Research Campaign until he contracted cancer himself. 'He'd raised millions, and when he was too ill to carry on, they sent him a signed photograph of the Duke of Devonshire. He was terribly upset.' He died when Rory was 18.
At the age of eight, Rory was sent to prep school, which he enjoyed; he is evidently not the kind of comedian who built a routine to avoid bullies. He was, he says 'a natural show-off. I have a need to be popular, which is both one of my greatest strengths and my greatest weaknesses.' He was also happy at his public school (Wellington, where he went on a scholarship) - 'the sporting facilities were great' - although now he seems to feel that perhaps he ought to have more reservations.
'I always felt I had a really good childhood, but it takes years for these things to sort themselves out in your mind. Compared to people who lived at home and played with their brothers and sisters, I don't know . . .' Nor is he sure whether he'd send his own children away, if he had any: 'Susie was a teacher for eight years in comprehensives and I respect her views on education. Things have moved on since we were children, and everyone sees the importance now of having family around.' Susie is his wife, from whom he separated earlier this year.
Bremner discovered a talent for voices by doing impressions of teachers. But his early ambitions were to be a sportsman, and his parents probably would have liked him to go into the army. 'My father was of the Portillo persuasion: I remember him being terribly proud when I stood up at the age of five and announced that the miners ought to be shot. It's taken me 25 years to realise it's the President of the Board of Trade who ought to be shot.'
In fact he studied French and German at King's College London, where he put out a satirical sheet in the German department, 'taking the piss out of lecturers and Goethe, full of literary references'. Towards the end of his time at university he was attending lectures during the day and performing in cabaret clubs in the evening. And then a show at the 1985 Edinburgh Festival led to an invitation from the BBC to present their best of the fringe programme, where he was spotted by Terry Wogan.
'I would have liked to be part of a group, but somehow it didn't happen,' he says wistfully of the alternative comedians of the Eighties with whom he started out, or the group behind The Day Today and Alan Partridge that is increasingly dominating the Nineties. Instead, he got sucked into light entertainment, with his own series: six shows a year between 1986 and 1992.
He didn't appear at the time to be struggling against constraints, although in retrospect he seems to think something about the BBC rendered him less savage. 'BBC production values are praised, and rightly so, but they can make things look very safe, so that everything looks like a BBC sketch show, shot with a slightly pinky filter with underlit steps. You feel you're practically wearing golfing trousers.'
IN 1989 Bremner went back to live performing. John Wells directed the show and encouraged him to be spikier, and he became dissatisfied with limitations he felt at the BBC. He left for Channel 4, which he feels 'exists in a way the BBC probably still should: trusting the audience's intelligence, not constantly wary that you're going above people's heads. You aren't asked to make so many compromises.'
Most importantly perhaps, Channel 4 gave him the freedom to record the night before transmission. He could listen to Major's conference speech on a Friday afternoon, and write his own version and record it that evening for transmission the following night. With John Langdon, his collaborator for 10 years, he writes 'two-and-a-half' sketches a week plus the monologues. John Bird and John Fortune improvise their sketches, and the producer, Geoff Atkinson, Sean Hardie, who worked on Not The Nine O'Clock News, and a 20-year-old woman, Debbie Barham, do the rest of the writing.
The regime when they are in production is punishing, and when Bremner separated from Susie, an artist who lives in Hampshire, they both blamed his work. He has since been seeing the GMTV presenter Penny Smith, but declines to answer questions about his private life: indeed, when they come up, he stops diving into different voices, blushes, and apologises.
The rest of the time he is constantly and slightly eerily metamorphosing into John Major, John Motson, Peter Snow and Michael Howard. Yet despite the almost automatic facility with which he does this, and the critical success of the last two series, he is not convinced he will do more (though he calls later to say that this was probably insecurity talking). He wants to produce, and plans 'something with Jo Brand next year', and he's hoping the production company will expand into documentaries. It would be a shame if he didn't take the satire further, because it is getting better. But that may be a minority view. The show is extremely popular with journalists, who can see how thoroughly researched and clever it is. But the BBC turned down his request to move from BBC 2 to BBC 1 because it thought he wouldn't deliver the ratings. He used to pull in four or five million viewers when he started at the BBC; now it's 2.6 million.
Bremner claims to be political 'in the sense that it frustrates me that a lot of people don't seem to be that bothered by what's going on any more', though he insists he isn't party political: 'I've been asked to help out on Luvvies for Labour, but I've made it clear I'm not up for that. I don't want to be fundraising for Screaming Lord Sutch one minute and taking the mickey out of him the next. My credibility depends on independence.' The current government is a gift to a satirist, of course; but even so, the protestations are not entirely convincing.