I think the "countdown" starts as soon as you receive the secondary regardless if you are verified or not. Most schools wait until you are verified. Most schools probably do not look at when they sent you the application and when you completed -- but some do. Some will also wonder if you are still interested in them if a month passes, and no response.
If you have the time though, fill out the secondaries as soon as you can. It will help you in the long run so that you don't have like 5+ secondaries to do all at the same time.
Also -- the schools that let you complete your secondaries before you are verified will be able to look at your application the moment you do get verified... instead of waiting another two weeks.
But really, there is no "countdown". Get secondaries done in a timeline that makes sense to you. If you want to wait until you are verified, than you can wait.