You're not screwed. Take a breathe. My girlfriend had the same as me, no patch, no play button on the launcher. Unfortunately she DID hit the install button.
I fixed my game due to this helpful topic:
http://forums.elderscrollsonline.com/discussion/comment/538476/#Comment_538476
What fixed my girlfriend's game was the following: Go to your game folder and rename your launcher folder and your Elder Scrolls Online EU folder to something else.
Close the launcher. Go to your account and get the link to a new client, click trough everything UNTIL you see the normal launcher with the game greyed out.
Stop, close the launcher again, rename back you game folder (the Elder Scrolls Online EU one) and use the launcher again, it will patch and work

-Tri
Edited by Triumph on April 7, 2014 7:33PM