With the resignation of Mrs May and the end of any prospect of Parliament passing the withdrawal agreement reached with the EU last November, it is looking very likely that the UK will leave the EU with no deal on 31 October. This is the default position under the EU Withdrawal Act 2018. Analysis by Maddy Thimont of the Institute of Government shows that the only way a no-deal exit could be stopped would be by Parliament passing a vote of no-confidence in the government. https://www.instituteforgovernment.org.uk/blog/new-prime-minister-intent-no-deal-brexit-cant-be-stopped-mps-0
The ‘Cooper’ clause added to the 2018 Act would only have effect in relation to any proposed ratification of the proposed withdrawal agreement with the EU. The clause in the 2018 Act requiring required the Government to hold a vote in the Commons if no agreement had been reached with the EU by 21 January is somewhat time expired now.
Who’d want to be PM now?