Do you still have both devices? Can you still connect the old device to the internet, such as via WiFi? Clash Royale (and Clash of Clans) has a build in tool, that lets you link to a new device, NOW it's usually used when you want to link from iOS to Android or vice versa, BUT maybe you can try iOS to iOS. I'm NOT sure if it will work. Go to Clash Royale on the old device, and then click on the gear icon next to your nick name on the main battle/arena screen, then click on Link Device, then This is the Old Device, THEN I want to Link to another Device, THEN follow the instructions.
Why can't you use Gamecenter?