The (payment) assessment date is the one that all other 'action' dates are determined by under NEC.
There is no obligation under NEC3 for you to submit an application for payment, although the contract may have been amended. Notwithstanding this, however, the assessment date remains as the last Friday of August.
The assessment date triggers the Contractor's obligation to certify within 2 weeks, which is the 'payment notice' under the 'Construction Act'. This certified amount is paid within 5 weeks of the assessment date which is 04 October 2019.
If the Contractor does not certify within 2 weeks, then your application for payment becomes the 'default payment notice' under the 'Construction Act' and the Contractor is obliged to pay this unless they give an 'effective' notice to pay less than this amount, which is required to be given by the prescribed period, which under NEC is 7 days before the final date for payment, see Y(UK)2 Y2.3, although this period of time may have been amended.
If an 'effective' 'pay less' notice has not been given then the Contractor is obliged to pay the amount of your application by 04 October. If that has not been done then you are also entitled to interest on the 'late' payment.
If the Contractor HAS certified within 2 weeks of the assessment date, which would be on or before 13 September 2019, this would still have to be paid by 04 October 2019.
Note that the final date for payment (5 weeks) is calculated from the assessment date, NOT from the date of the certificate.
Hope this helps.