If the TEST button isn't firing the flash, then something's probably wrong internally with the flash and it needs to be repaired (e.g. bulb is blown or the capacitor isn't working). I don't think there's anything you can do to fix it if you don't have electronics/soldering repair knowledge and tools. If it's still within the one-year warranty period, I'd go back to the seller you purchased it from to request an exchange. If they won't do it, and you bought from Amazon, you can try them or your credit card company.

Generally, we tell folks to purchase the Adorama Flashpoint R2-rebranded versions of Godox gear, because they're known to supply warranty and customer support.

If you need something fast, the AA-powered TT600 is only $65 and can do most of what the V850 III does (though not ID codes) and with an older different UI.