July 23rd - new Tory leader
July 25th - Summer recess
Aug 1st - Radnor by election
Sept 3rd - Parliament opens
Sept 14th - Party conferences
Oct 7th Parliament re-opens
Oct 17th - European Council
Oct 31st - Brexit extension expires
If they call it on September 3rd after recess, then an election on October 3rd
If they call it on October 7th straight after conference season, then an election on October 31st
I think the party conferences follow summer recess, and the state opening of parliament follows the party conferences.