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It all began with a Miss Charlton, who mixed up her lines in September 1979. “About the beer?” she asked host Robin Day at the start of the first Question Time. “Oh, I’m doing the wrong thing, aren’t I?” she added, fiddling in her handbag — before Day stepped in to rescue her: “I’ll tell you what it was: what do the team like doing on a night out?”

After that wobbly start in front of a panel including the then Labour leader Michael Foot, a Right-wing Tory, a bishop and a token woman, no one would have guessed that 40 years on the show is taken seriously enough that the choice of its next presenter has something close to constitutional significance. For the lighthearted start to the show soon gave way to a seriousness that has made it the definitive choice for politicians to communicate with the public.

Today, applications close in the search for a replacement for David Dimbleby, the third regular presenter to hold the role — there have been twice as many Prime Ministers in that time. While many of the usual suspects are being considered for the plum post, BBC insiders say that there is growing panic at rumours that Piers Morgan is gunning hard for the job. It would be an audacious choice: it’s undeniable that Morgan would make compelling viewing, but the question is whether his appointment would prove too disruptive for the Beeb.

Other shows have majored on tough interviewers but only Question Time has them chairing a panel of rivals, before a noisy crowd. I’ve helped to prepare people for the show and tried to cheer them up after staggering off it. If you thrive, your political share price shoots up. Fail, and everyone knows about it.

There are three things people say about Question Time: that it’s got too raucous, that its panel isn’t balanced (too much Nigel Farage , too many air-headed celebrities, too few scientists) and that no one watches it any more.

The latter definitely isn’t true. Even if the programme doesn’t make the top 30 most-watched on BBC One (for real national fame, appear on Countryfile), it shapes debate. And that’s the BBC’s big challenge. Should it change a winning formula or just try to keep an old one fresh?

Some of the guests on the first few programmes four decades ago — former Labour leader Neil Kinnock, for example — are among those still being invited on today. But Britain is a different place.

"The BBC will get complaints if it chooses a host from the old mould - the pressure to pick a woman is irresistible"

Day set the pattern for hosts which seems dated today: male, established, opinionated and white. All three of Question Time’s — Day, Peter Sissons and David Dimbleby, who has clocked up 25 years — have been brilliant at the job. The trick isn’t just to be the “highly respected journalist” and “figure of authority” required in the BBC’s five-page internal advert for Dimbleby’s successor but someone who can bring the public into that conversation.

That’s why, when the BBC last tried to fill the post and pitted Jeremy Paxman against Dimbleby in tests, the latter won hands down. You need warmth, not just incisive contempt.

On air, you have to be able to handle a crowd — and you have to be fair. They’ll be newcomers to the show as well as old hands. Political neutrals as well as party bruisers. Cabinet ministers (the best, anyway) know their stuff inside out but it’s your job to jump in with a correction: “… but minister, that’s not what you promised six months ago” — and then nail that point right away.

Dimbleby is a past master at it, which is why people watch and why they want to appear on the show. If the BBC gets his successor wrong, it will soon know about it. But it will get complaints, too, if it just chooses a figure from the same old mould.

So, yes, if it wants to play safe, Andrew Neil is the obvious choice. He’d terrify politicians but he’d come over as the audience’s friend. He knows his stuff twice over. He’s the continuity candidate and, for a programme which in many ways works, there’s plenty to be said for continuity.

But then some things should change. Maybe the show, like politics, has got too divisive — more mob rule than heated discussion? It’s impossible to imagine the host’s job being described as “a lighter assignment”, which is what The Times said on Day’s arrival in 1979. Is it becoming the TV equivalent of Twitter— just too angry?

In which case, there’s no shortage of other options on who might keep a grip but change the tone. Newsnight’s Kirsty Wark says she wants the job and there’s no doubt she’d thrive. She’s surely a failsafe frontrunner. The BBC’s Nick Robinson would be a strong contender — but he’s a man, political to his core and the pressure to pick a woman fourth-time running must be close to irresistible.

According to Nick Pisani, who’s had his fair share of worrying about absent guests on late trains as editor of the show, there are five tips for getting the job: don’t be a man, enjoy dealing with the public, command respect from the panel, have a sense of humour and be on top of your brief. That might point to another frontrunner, Emily Maitlis — with Wark perhaps getting the top slot at last on a revamped Newsnight.

These talents might sound a bit familiar, though. If so, what about Mishal Husain, who is friendly, sharp and steered her way impressively through the egos of some of her fellow Today presenters (“It’s not pretence, we really don’t like each other,” one of them once told me). The BBC’s media editor Amol Rajan has also been mentioned — his switch from editor of The Independent to a TV personality has gone smoothly.

Radio 4 presenter Carolyn Quinn’s talents are surely bigger than the shows she’s been given so far. And then there’s Samira Ahmed, who presents Radio 4’s Front Row and lobbed in her application on Twitter. Good though they are, these names are still terribly BBC. And maybe bosses will go for a high-voltage shake-up, to keep it a must-watch — at least for those who like a scrap on screen.

If so, perhaps it will be Morgan after all who gets the call. The political classes might recoil in horror — but then if any show is designed to make them do that, then surely Question Time is it.

Runners, riders and rank outsiders

Andrew Neil 30/1

Strong form like his can’t be discounted. He’s the most acute and persistent political interviewer out there. A safe choice for BBC bosses if they go with a man — but they’d have to be prepared to face down criticism over his supposed views on Brexit.

Kirsty Wark 5/2

The frontrunner — no one doubts that she’d handle the guests, though keeping the audience eager as well as angry would be a new test. But leading from the front in any race is difficult, and she could just be overtaken on the last circuit.

Emily Maitlis 6/1

At home across all sorts of TV ground, she’s got what it takes to become the next Dimbleby. Less favoured so far in the stakes but don’t rule out an impressive run-in to take the prize.

Mishal Husain 9/1

Today would be sorry to lose her but Husain might be the right pick if the BBC wants to steady the shouting on a show where public passions risk driving the calm centre away. On the right day, an outside tip to win.

Piers Morgan 50/1

He wouldn’t just present the show, he’d be the show. But isn’t that what they said about Robin Day? If you want someone to pick up politicians with the one question they hadn’t prepared for, he’s got form. But maybe if he’s in the chair certain politicians will simply stop coming on?

Samira Ahmed 18/1

Could the tweet work? A good performer on all sorts of ground. Might falter on a difficult night in Doncaster — and other candidates would ask what she has that they don’t.