UART connection between Arduino Nano and NodeMCU
You can't connect the same Serial to a computer over USB and to a Nano at the same time. Not only because the Nano is on 5 V which disturbs the 3.3 V side of the USB converter. 

Disconnect the NodeMCU from computer, try to power it from Nano's 3.3 V pin (not for WiFI) and don't forget to connect ground between the MCUs.

Nano has one Serial, which is connected over USB to computer too. So there too you can't use the USB and RX TX pins at the same time. You could use SoftwareSerial to connect to NodeMCU.

The Nano TX to esp8266 RX should have logic level conversion to not to send 5V to esp8266 3.3V pin.