The veteran comic was one of the Prime Minister's favourite comedians. He and his team from The News Huddlines, a unique combination of topical satire and music-hall variety sketches, have scripted jokes for both Blair and Prescott when in opposition. No longer.
When I went "backstage" after watching Hudd record an episode of the world's longest-running radio comedy in front of an audience, he told me his love affair was over. One of Britain's best-loved comics had sent one of Britain's least-loved spin doctors away with a flea in his ear. "Alastair Campbell has rang us up," said Hudd, "and asked for more jokes. But I told him, 'I'm sorry, but our writers don't like your government.' They haven't asked again."
At one Labour conference in opposition Hudd gave Blair a topical joke about Eric Cantona kicking a football supporter, with Blair saying, "Doesn't he know it's the Conservative government's job to kick people in the teeth? Blair's delivery left Hudd unimpressed. "He even buggered that one up," Hudd recalls forlornly.
But then few Prime Ministers could impress a man who is an expert on music hall, and in his early days performed with members of The Crazy Gang. And Blair might have been less than comfortable if he had attended this recording of The News Huddlines, which included a sketch about Rupert Murdoch's marriage breaking up because Mrs Murdoch found Rupert in bed with ... Tony Blair.
The audience may be older and less irreverent than their equivalent on Have I Got News For You (one gentleman, sporting a British Music Hall Society badge, who has been coming since the programme started in 1976, told me he attended every show "except when I'm in hospital") and the gags are more gentle, but the target of many of the jokes is the Government, and the aim is sure.
The evergreen June Whitfield does a simpering, lisping, Harriet Harman which deserves a wider audience. Health minister Frank Dobson's line of "I'm very, very sorry" for refusing to give the nurses a pay rise is turned back on him in a sketch when doctors are "very, very sorry" they do not have money to complete his operation.
And for music-hall lovers in the regular audience, there are constant allusions to the genre by the talented team of writers.
The six of them gather a couple of days before the recording. Each has two hours to come up with a sketch based on the week's news. One writer, Glenn Mitchell, who has written books on Charlie Chaplin and Laurel and Hardy, came up with a limerick about the woman who had a Teletubby tattooed on her breast to encourage her baby to be breastfed. She didn't have room for the cast of Blue Peter, was the gist of the limerick. But a few in the audience recognised it was written in the style of the old music-hall star Billy Bennett.
The show is an odd hybrid: conventional gags (a news item about honeymoons on the moon: "You can seen the earth move even if you can't feel it"), mixed with dollops of political incorrectness (Hudd does Trevor McDonald with an exaggerated West Indian accent performing a news calypso).
A further unique aspect of the show is that it is the one remaining broadcast comedy to which aspiring gag writers among the public can contribute. And if your joke or sketch is used, your name is read out at the end. The News Huddlines has been a training ground for the best comedy writers and producers.
They may have even got material from studying the Huddlines' regular audience. Hudd's press agent, Laurie Bellew, who has represented Peter Sellers and Ken Dodd and is a walking encyclopaedia on comedy, recalls deadpan that for years three civil servants, two men and a woman, would come to the recordings. For the first half of the show, the lady would snog one of the gents; for the second half, she would snog the other. At the end, the three would pick up their briefcases and proceed respectably back to Whitehall.
Back in the studio for the latest recording, Hudd and the team have to re-do one gag where the sound was slightly wrong. It was a joke based on a news item about AA and RAC "passion patrols" finding courting couples when they attended breakdowns. "How did you know it was us who needed you?" ask the couple. "Because yours is the only car with the windscreen all steamed up," says Hudd's patrolman. A pretty average joke. And it seems Hudd knows it.
They do a re-take. "How did you know it was us?" "Because yours is the only car with footprints on the inside of the windscreen."
The ad-lib sends the rest of the cast and the audience into delights of laughter, and it is included in the eventual broadcast.
That's the mark of a genuine veteran comic. But I doubt Tony Blair will be using the joke.