Hello @felmue
I got it working now. The PWRKEY pin needs to be low for a minimum period of 1 second and a maximum period of 12.6 seconds. After this the PWRKEY pin needs to be put high again, otherwise unpredictible things might happen. Subsequently there is a minimum time defined in which the UART is ready. This minimum time equels 1.8 seconds after power-on. (table 8, power on timing and electronic characteristic).
So my final (working) code is:
pinMode(GPIO_NUM_27, GPIO_MODE_OUTPUT);
digitalWrite(GPIO_NUM_27, HIGH);
delay(1000);
digitalWrite(GPIO_NUM_27, LOW);
delay(800);