What I understand from your question is somehow template is not detecting changes or it maybe that you are not changing the correct index.
You can try these:
check if you are changing the correct index of the array if yes then :
try reassigning the same variable which generally triggers the change detection mechanism like this:
this.negotiationRounds[roundIndex][1].quoteCoverageAssessments[quoteSupplierIndex] = {...assesment}
or you can manually trigger the change detection mechanism by importing the changeDetectorRef and triggering the detectChanges method:
this.changeDetectorRef.detectChanges();