It is the Tories who are facing a leadership election this morning – but who could replace Theresa May?

Before the election, the only MPs who were talking about their party’s leadership were Labour. One told me that if Jeremy Corbyn were still leader by Christmas, they would be “off”, sitting in Parliament as an independent, and that there were plenty of others who would do so too.

Labour MPs speculated about whether Tom Watson could persuade Yvette Cooper to hold back from launching a leadership challenge Friday – Watson’s view being, apparently, that it would be best to delay. The deputy leader was said to want to change the party rules to bring back the electoral college to give the MPs a bigger say in the choice of leader.

Instead, this morning it is Conservative MPs who are hurriedly consulting each other about names and the timetable. Theresa May has to go, they all know that. But who will replace her, and when?

The first instinct of most MPs is to take their time. They would rather have the full leadership contest that they failed to have last time; many of them think it was a mistake not to put May and Leadsom through the test of hustings in front of party members in the country. Many of them are bitterly critical of May’s lack of campaigning experience, which they think was exposed in the general election campaign.

The problem is that, as May repeatedly said during that campaign, the Brexit talks in Brussels start in 11 days’ time. It seems that officials will have to start those talks under a caretaker prime minister, unsure about what will happen to their negotiating mandate when she is replaced. After the British Government’s preparation for Brexit talks was delayed for seven weeks because of the general election, it will now be distracted by several more weeks of leadership election.

Then there is the question of who the new prime minister will be. Boris Johnson, David Davis, Amber Rudd and Philip Hammond are the obvious candidates. Rudd, having only just held her seat at Hastings after two recounts, might seem a little precarious; if she were the right choice that shouldn’t matter.

The question that Conservative MPs have to ask is whether someone who was an out-and-out Remainer could negotiate Brexit, as opposed to May, who was a reluctant Remainer. And that depends on what effect they think this election has on the kind of Brexit they should be trying to negotiate.

Just as May interpreted the EU referendum to mean a particular kind of Brexit – ending the free movement of people – could a new prime minister interpret this election as a hesitation about the rigours of a hard Brexit?

Rudd had a good election, but she is hardly a seasoned campaigner with a following among the voters. Yet the obvious candidate in that category, Johnson, arrives trailing a cloud of doubts and cavils. He showed that – unlike May – he really could appeal to Labour voters as Mayor of London. He is not as popular as he once was, however, having hardly put a foot right as Foreign Secretary.

Davis has been the most successful of the trio of Brexit ministers, surprisingly pragmatic and consensual in his approach, but is he really the leader to broaden the party’s appeal? Then there is Hammond, who at 21:59 yesterday looked as if he might be about to lose his job, and who might now be a contender, as the safe pair of hands candidate, for the one more senior post in government. Again, though, if May tried and failed to widen the Tory party’s appeal from its supposedly narrow base under the Cameron-Osborne posh boys, why would Hammond succeed?

If I were a Conservative MP, I would be looking admiringly at Ruth Davidson, who has led the Scottish Conservatives to such as stunning recovery. I would note that she is a Member of the Scottish Parliament and not of the House of Commons. And I would be suggesting that, when there is a by-election in Maidenhead, the Conservative Association there should look to Scotland for its candidate.