Depends on whether they do a motion of no confidence first and it passes.
Bercow. What do you reckon he'll do? Go for an extension to prevent no deal. Or go all out to revoke article 50 or perhaps a second referendum. Timescales are not going to be great with the summer break and party conferences so I reckon he'll only get one shot. Extension is guaranteed to be passed. Not so the other 2 so we could still end up with no deal brexit come 31 October.
I'm not entirely sure that a motion of no confidence would need to be passed first. If the new Tory leader decides to dig their heels in and go for the legal default "leave with a no-deal" on Oct 31 then Bercow has said that Parliament must be allowed to debate and vote on this position.
So first-up, I would expect him to initiate a vote on a no-deal brexit. If the result of that vote, is parliament rejecting a no-deal and the Tory Leader still decides to push on for a no-deal then the motion of no confidence would probably be raised. But I think that motion would need to come from the main opposition (labour party) and not the speaker?
Then if that motion is passed, we are into a GE situation. Then and only then would we need to approach the EU for an extension of the Oct 31 deadline.
TBH I haven't got a clue how this will turn out - anything could happen
