Problems whit external power nano need physical restart on board, before schets running. USB to computer, no problems...
Because the RX pin is unconnected, the bootloader on the board may be seeing garbage data coming in, meaning that it never times out and starts your sketch. Try tying the RX pin to ground with a 10K resistor (or connecting RX directly to the TX pin).
Tested, not help..
Thanks for your time.