@user2684 It appears that this problem is caused by a bug in bootloaders used in some Arduino boards.

The solution suggested is to flash optiboot bootloader to the Arduino Pro Mini. Or not using a bootloader.

I use mostly barebone atmega328's with optiboot bootloader, working perfectly with reboot function, so I haven't been aware of the problem until now when I used a pro mini.

Please read this article which explains the cause of the problem and suggested fix.
https://andreasrohner.at/posts/Electronics/How-to-make-the-Watchdog-Timer-work-on-an-Arduino-Pro-Mini-by-replacing-the-bootloader/

I will try it on a pro mini when I have time.

Hi, nice work!
I have changed the code for REBOOT so there is no need for a reboot-pin.
like this...

else if (strcmp(message, "REBOOT") == 0) {
#if DEBUG == 1
Serial.println(F("REBOOT"));
#endif
// Software reboot with watchdog timer
// Enter Watchdog Configuration mode:
WDTCSR |= (1<<WDCE) | (1<<WDE);
// Reset enable
WDTCSR= (1<<WDE);
// Infinite loop until watchdog reset after 16 ms
while(true){}
}