You need to call Apple directly and ask for AppleCare Administration.
This happened to us as well. The first phone (of two) we ordered, is already in service and we did not receive the AppleCare plan email. The other phone, which is being shipped, the email arrived already.
I knew something wasn't right with that. And it turns out it wasn't right. AppleCare had no record of me purchasing the plan and I had to actually send them proof. They were very nice about it and I now have the plan details for both phones. But had I not called them, the plan would have never shown on the phone we just got Friday.
It will take about 72 hours for the plan to appear in the Settings > General > About.
But based on what you've written, I recommend you call them.