Can someone clarify the upgrade process for me?

So during the new event, we have to wait (or pay GC) for the time to "analyze" the upgrades.  Then once this time is over we can purchase the upgrades using R$ and/or GC.  Do we have to wait again for the upgrades to be ready after we purchase them?  Also when does the "analysis" start, after you complete the event Eric talks about or when you first get to that stage?  

Paul