That sounds to me like you didn't connect the ground / 0V between the Nextion and the Mega when you were using 2 different power supplies. No, they don't need to start at the same time, well, not with my code anyway.
I can't comment without more details, which means knowing exactly how you had it wired (schematic, not list of this connected to that) YOUR code exactly as you had it (code tags please) and your HMI file.