No response in Serial Monitor from es8266 with Arduino nano
If it is not a genuine Arduino (Chinese clone probably) and it has the CH340 USB to Serial converter chip, then it will not work as the digital Signal on TX does not get pulled to ground when transmitting a digital LOW, (if that's the right way to say it). There seem to be resistors on the board that cause this effect, or something. I do not guarantee that this is your problem but it's highly likely.
Use an FTDI USB to Serial board or similar and you might have better results.