the Uno WiFi Serial1 library is in Library Manager in IDE. better install it there, not from GitHub.
please do not call this library firmware. we call a 'firmware' the firmware for the esp8266. the Uno WiFi Serial1 library is only a gate to esp8266.
the EspProxy sketch from Uno WiFi Serial1 examples has two modes. an automatic mode and a hard-to-bootloader mode. the settings is the #define FLASHING at the beginning of the sketch.
for WiFi Link firmware installation we need the hard-to-bootloader mode. warning in this mode the esp8266 doesn't start the user program, (it is io 0 LOW all the time). so you need to remove it at the end.
so after installing the Serial1 library, open the EspProxy sketch, uncomment FLASHING and upload the sketch into Atmega.
the next step is to erase the flash of the esp8266 to clear all old SDK settings. do you have esptool installed. if yes, run esptool.py -p COMx erase_flash