If the opposition wins a No Confidence vote in the week of April 8, 2019, could they stop No Deal?

Question

No Deal Brexit is currently on schedule for April 12, 2019. Meanwhile, the news reports this week (April 3, 2019) that various Conservatives are considering voting with the opposition in a No Confidence vote to bring down their current government under Prime Minister May.

Normally, opposition leader Corbyn would welcome such news if it were true. He called a No Confidence vote in January, but the government won. If some Conservatives were expected to vote with the opposition, he could call another No Confidence vote soon.

However, I'm wondering if this would be a Sophie's choice* for Corbyn, in which even if the opposition won a No Confidence vote, it could make No Deal the inevitable outcome of Brexit on April 12? Corbyn has previously been adamant that No Deal is a horrible outcome and should not be considered an option. The problem would be in the mechanics: The obvious ways of stopping or delaying No Deal all depend on executive action---either for the prime minister to revoke Article 50, or request an extension. The EU has clearly stated that it can only negotiate with the executive, not directly with parliament. But if the opposition wins a No Confidence vote, it would take a while to establish a new executive, longer than April 12.

In such a scenario, who would hold executive power in the interim? Would parliament continue to function? And would there be any way for parliament to stop No Deal Brexit from happening?

_{*EDIT: or Pyrrhic victory. Calling a vote would be the choice. Winning it would be the victory. Sorry for the mismanaged political metaphors...}

Answer 1

I think the following sequence is possible:

• No-confidence motion in the FTPA (Fixed Term Parliaments Act) form to collapse the government. This does not automatically trigger elections immediately.
• Immediate confidence motion in someone else - not necessarily Jeremy Corbyn! It is conceivable that someone, even a member of the House of Lords, could be chosen as a "caretaker" PM for this purpose. This would rely on getting 320 MPs who are currently fighting like cats in a bag to agree on a choice.
• Caretaker PM announces revocation of Article 50 and indefinite deferment of the coming into force of the Exiting the EU Act.
• Caretaker PM resigns (I think due to the operation of FTPA this requires a no-confidence vote even if they resign by choice?)

With appropriate organisation it could be done in an afternoon, with the candidate driven to the Palace and back in the middle. The Australian Constitutional Crisis managed it in a day.

Answer 2

When a general election is called, the (potentially) outgoing PM is still PM, and Parliament is dissolved. May would continue to be PM until the new Parliament appoints someone else, after the election is resolved (or they keep her).

Traditionally PMs don't plan to do anything controversial during this time, as they are focused on fighting the election.

But because of the Fixed Term Parliament Act, the only way she can call such an election is with the agreement of a majority of Parliament. To avoid a no deal, Parliament could instruct May to ask the EU for a further extension, giving enough time for an election to be fought, and only if that is granted vote for an early election.