Shared GND? It needs it.
Otherwise I do not know. I run a Mega/ESP-01 combo and they both power up together.
Powering the ESP-01 first, then the Mega works fine.
But if I power the Mega first, then obviously it gives the error as the ESP-01 is not powered, as is needed in order to respond But I have a auto re-connection routine that handles that anyhow and thus it will work as soon as the ESP-01 is powered up.