If your board gets reset by the watchdog, the watchdog does NOT get turned off by a reset, which means it is still running in the bootloader. The timeout DOES get reset to the shortest possible time, however. So the bootloader has to disable the watchdog, or you’ll go into a reset loop while the bootloader is checking for input.
What board are you using?
There was a bug in some older versions of the bootloader, where it did NOT disable the WDT, leading to the sort of behavior you’re seeing. If you’re using a Nano “clone” with the old bootloader, this is probably what you’re seeing.
See also: ATMEGA328 locks up on watchdog reset