Replaced iPhone 8 Battery, Now Home Button is Dead
You have a couple options.  For diagnosing the issue you could buy a new display to see if that is the issue, or buy a new home button called the JC home button.  It will not restore Touch ID, but it will tell you if the screen cables are broken, or the home button itself.
Before you do any of that, try using the old battery again to see if your home button works again.  Also take off the display and reseat the display cables.
I have replaced several batteries on iPhones and have never had that issue, so I'm guessing a cable isn't seated correctly or a ribbon wire got torn.