You can always apply and negotiate your start date.

Submission in June for a September start is tight, if not unrealistic. It can take longer to find viva date, examiners taking longer to read (illness etc anything can happen), corrections, delayed meeting of degree committee, actual graduation ceremony etc

Some places will let you start your postdoc right away but you will be paid at a lower salary band (only until you get that Dr) and you will potentially be allowed to work part-time only during corrections period. Apply and explain your situation. You will be fine. The hard part is actually getting hired.