I have AT&T pre-paid and the automatic transfer didn't work for me. I called AT&T (611) and I was on the phone for 3 hours, literally, not exaggerating. The rep was very nice and we kept at it, we tried probably 6 different esims before it finally stuck. He asked me to keep restarting the phone, etc. and to turn off the wifi after it started activating. The problem was, it would just continue to try to activate without actually activating. Anyway, the trick is I believe to make sure you forget your wifi network completely after it starts activating and to restart the phone (once you restart your phone, it'll reconnect automatically to wifi). Once it reboots, it's good! I spent 2:45 on my phone and it only took 15 minutes to do my wife's phone with this method. The AT&T rep believes it's a bug with the software.
So if you have to call AT&T and you give him the your iPhones information, the pop up will show in the settings, click activate, it'll activate and it'll say "connecting to network" then after a couple minutes it will tell you that you can continue to use your phone while it activates. Forget your wifi network, and restart the phone. Hopefully that will work for anyone who is running into this problem with AT&T