The Godox TT520 II is typically sold with the Godox 433 MHz RT-16 transmitter. The TT520 II and the RT-16 transmitter are both single-pin devices, so the only thing a flash hotshoe can tell the transmitter and the only thing the transmitter can tell the flash is when to fire.
The first thing you need to check is that the 5D Mark III's flash menus/settings aren't set to use the Canon wireless settings, TTL, HSS, or second curtain. None of those things can be communicated to the transmitter or the flash. The transmitter needs to be treated as if it were an on-camera flash, not a wireless master. Canon's wireless protocol is completely different and separate from what the Godox gear is doing.
Secondly, make sure the TT520II is set to be a radio receiver. The user manual says to make sure it's in radio receiver mode, you need to hold down the MODE button for two seconds until the LED with the antenna icon lights up.
Thirdly, make sure the four little DIP switches for setting the channel on both the transmitter and the flash match so they're on the same channel:
You may also want to check that S1 and S2 optical slave modes are turned off, and that the 23A battery in the transmitter is still good.