Theresa May has survived an attempt to change Tory party rules to make it possible to oust her within weeks, but backbench MPs have demanded a “roadmap” for her departure, in a fresh blow to her authority.

Amid a growing sense of political paralysis at Westminster, Brexit-backing members of the executive of the powerful 1922 Committee were keen to change the rules that insulate the prime minister from a fresh leadership challenge until December.

After a second meeting in two days, the committee decided on Wednesday to leave the rules unchanged. However, the 1922 Committee’s chair, Sir Graham Brady, said MPs would like the prime minister to set out her plans for stepping aside, if parliament failed to back a Brexit deal in the coming weeks.

“Following the prime minister’s decision to set out a schedule for her departure as leader of the party should the withdrawal agreement pass, we seek similar clarity from her in other circumstances,” Brady told reporters outside the meeting. “We should have a clear roadmap forward.”

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May had already pledged to quit almost immediately if the withdrawal agreement won a majority, though that promise failed to win over enough Tory critics to prevent the deal being rejected by MPs for a third time last month.

Before the Easter recess, the prime minister urged MPs to “reflect on the decisions that will have to be made swiftly on our return”. But with little sign of progress in ongoing negotiations with Labour, it is unclear when the government may feel it can test the support of parliament again by tabling the withdrawal agreement implementation bill (WAB).

The shadow foreign secretary, Emily Thornberry, renewed Labour’s criticism of the government on Wednesday as she urged the Cabinet Office minister, David Lidington, May’s de facto deputy, to make a commitment to compromise in the talks by agreeing to adopt a customs union.

But Lidington pointed to the advantages of the prime minister’s deal, which he said was aimed at securing the “benefits” of a customs union, and argued that the customs union was one of a string of other options that had already been rejected by MPs.

“I think that there is a genuine attempt to find a way through. However, I will not hide the fact that this is very difficult, because if it is going to work it will mean both parties making compromises and our ending up with a solution which, unlike any other proposed so far, will secure a majority in the house,” he said.

“So far, the house has rejected our deal; it has rejected the opposition’s proposals; it has rejected a referendum; it has rejected revocation; it has rejected a customs union; and it has rejected common market 2.0,” he said.

The chancellor, Philip Hammond, met the shadow chancellor, John McDonnell, on Wednesday to discuss financial services after Brexit, as part of the ongoing dialogue between the two sides.

Another session, on environmental policy, will take place on Thursday, but no date has been set to reconvene the wider group – which includes Lidington and other key figures such as the Brexit secretary, Steve Barclay – and both sides appear frustrated about the lack of concrete progress.

Downing Street said there were no plans for May to respond imminently to the 1922 Committee’s call for more clarity about her departure plans, saying she had just returned from Belfast.

Earlier, a spokesman declined to say whether the Conservative party would hold a campaign launch for the European parliamentary elections. Formally, the government’s position remains that it hopes to secure parliament’s backing for a Brexit deal in time to leave the EU next month, and avoid participating in the elections.

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But with time running out, few at Westminster believe that will be possible and the prospect of fighting elections against Nigel Farage’s Brexit party is infuriating Tory MPs.

The prime minister saw off a no-confidence vote at the end of last year and the party’s rules stipulate that a fresh challenge cannot be launched for 12 months after that date.

Brady said the decision by the committee on Wednesday was the result of “two lengthy meetings and a good, full and constructive debate, in a friendly and collegiate way”. He said he would not disclose how many dissenting voices had been heard in the room.

During the full meeting of the 1922 Committee that followed the executive’s decision, Tory MPs urged some of their more intransigent colleagues to back the prime minister’s deal and swerve the EU elections.

One Brexiter, Andrew Bridgen, stood up to demand that the whole parliamentary party should be given a vote on the rule change, but was rebuffed by the former education secretary Nicky Morgan, who likened it to backing a second referendum. “That seems to me to be very much like re-running a vote that happened six months ago,” she said.