I bought a Lumia 1520.3 from B&H and I cut my sim card down from a micro sim to a nano sim. I know it still works because I put it back in my 920. It says on my 1520.3 says no service in the live tile.
If you swapped the SIM in or out of the 1520 while it was on, that may be your issue. That, and cut SIM cards don't work well in general. The adapter you're using to put it in the 920 may be why it works in it. Try getting a real nano
Did the nano sim resolve the issue? I just got a 1520 that can't connect to a network. I have a nano sim and I checked it in another phone and it is working.
Last time something like this happened to me. My carrier was T-Mobile. I took it in; they did nothing special just switched the SIM card, and they fixed the problem.