I got the address check email from an individual and then an hour later that same individual emailed me and said they had processed my application and added my name to the Foreign Births Register. The individual in the second second email said that my certificate was in the queue to be printed and this was currently taking a number of weeks. I received a tracking number at the same time.
About a month later I got the form "congratulations" email. And a few days after that, the tracking number was on An Post - it's currently en route.