ROBERT SIEGEL, HOST:
And you thought our politics were chaotic. Consider Britain's two main political parties. Their leaders, the Conservative prime minister and the Labour party's leader of the opposition, both supported staying in the European Union. Prime Minister David Cameron is now set to be replaced in September.
Labour leader Jeremy Corbyn lost a no-confidence vote among Labour members of Parliament that was not close - 172 to 40. And he insists on staying. In the House of Commons today, Cameron was blunt with his opposite number.
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DAVID CAMERON: It might be my party's interest for him to sit there. It's not in the national interest, and I would say, for heaven's sake, Man, go.
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SIEGEL: Well, to help us sort out the post-Brexit referendum political comings and goings, we turn to John Peet, political editor of The Economist. Welcome to the program.
JOHN PEET: Hello.
SIEGEL: First the Conservatives - if it's not David Cameron in number 10 Downing Street, who will it be come October?
PEET: I think the favorite continues to be Boris Johnson, the former mayor of London who was one of the leaders of the campaign to leave the European Union, the Brexit campaign. And I think he probably still is a favorite. But he's not very popular with quite a lot of MPs, and he's obviously not very popular with those who wanted to stay in the European Union.
So I'd still say it's a bit close to call. Theresa May, the home secretary who did back remain but is a known Euroskeptic, will run. And there are at least two or maybe three others. So by the end of the week, I think we will probably have something like five or six candidate for the leadership of the Conservative Party.
SIEGEL: And who will decide which one of them will be the prime minister?
PEET: Well, it's a complicated system. I mean, it's not a primary in exactly an American way. The answer is that Conservative members of parliament vote in a secret ballot on all the candidates, and they winnow it down in successive ballots until there are two of them - the top two, as it were.
And the top two candidates then go out to the members of the Conservative Party nationally in a ballot that will probably be held during August or early September. And by early September, one of those candidates will have been chosen by the members of the party.
SIEGEL: Now, let's turn to the Labour Party. Jeremy Corbyn - obviously very unpopular with Labour members of parliament, but he was chosen by the members of the party at large and thinks they still support him. If not Jeremy Corbyn, who? Who would lead the Labour Party?
PEET: Well, you know, I would love to know the answer to that question (laughter). I mean, we have a very plausible candidate who declared today - Angela Eagle, who has acted as Jeremy Corbyn's (unintelligible) prime minister's questions. And she's quite popular based in the party and with Labour MPs.
But the problem with the electoral system here in both parties but perhaps even more in the Labour Party is that in the end, it will still go back to the membership of the party, which is considerably more left wing than Labour MPs.
And Jeremy Corbyn is still insisting that if it comes to a leadership election, he will run again, and he will probably win again. And the Labour MPs could then be stuck in the same position as before with a leader They don't support thanks to the members of the party.
SIEGEL: Is this an unusual time for so much flux in the leadership of both the Conservative and Labour parties in the U.K.?
PEET: I think it's the most unusual time in my lifetime, so I guess, yes, (laughter) you could say that. You know, we've had some quite difficult times at different periods when the economy was in trouble. Harold Wilson had trouble in the 1970s.
But the fact that both parties are pretty rudderless and that we've just had a referendum which has produced a result that most MPs in both parties disagree with, we're in completely unchartered political waters. And nobody really knows how this is going to work out in the next few months, let alone the next few years.
SIEGEL: John Peet, political editor of The Economist, thanks for talking with us.
PEET: Thank you. Transcript provided by NPR, Copyright NPR.