Home button not working after screen and battery replacement?

It’s incredibly easy to break the home button when doing a repair on an iPhone if you don’t have any prior experience working on them. It sounds like you damaged the home button during the repair. You may have been to rough when you where taking the home button off the old screen or you may have pinched the cable when you where putting it back together. It’s possible that you only damaged the home button extender cable that’s attached to new screen. Unfortunately that cable is part of the screen so you can’t just try a new one. You would have to try a whole new screen. You have to be super careful where the cables and connectors get seated as even a little pressure can cause major issues. Worst case scenario, you have two options. Either replace the home button with a new one and accept that fact that Touch ID will never work again or take it to an Apple store and have then replace the home button for you. Apple is the only one that can link a new home button to your phone so that Touch ID will work with a replacement home button.