I'm using a physical keyboard on the nothing phone and I don't have any input lag problem. I am using it mainly for termux admittedly but I haven't perceived any in other apps.
I just tested it out and it didn't work well for me. It reliably started either on the current app or on the latest app when on the home screen but I haven't managed to change windows at all. That could be due to my weird keyboard though. It only has 34 keys and has a weird way to reach the alt key which might be messing with everything.
I'll see if I can try with a standard keyboard later.