Home button is not working after new screen replacement

There could be several reasons why this occurs:

• The flexcable between the actual button and the motherboard has been damaged during the repair, or is not properly connected.
• The homebutton has a backplate to ensure it clicks properly, and also registers the clicks. If this isn't properly mounted, or completely missing, it could very well be a reason. You can easily tell if that's the case if you click on the button and you feel like there's nothing these and the button just "sinking in".
• The homebutton itself is damaged and needs to be replaced.

Those above would be the most common issues as far as I'm concerned, however it's not guaranteed that it's one of them.

Troubleshooting

• Examine the flexcable carefully and search for any potential damage. If you can confirm, for example, a crack, that's most likely the issue. To fix that, you may need to order a new flexcable, or an entirely new button.* You should also ensure that it's properly connected and mounted, as that could also be your issue.
• Examine the home button backplate. Look for any missing screws or any other damage. As I said earlier, you could easily tell if this is the case by clicking the home button, and if it feels like you're pressing on nothing, you have your potential issue right there.
• If the home button itself is damaged, it would be a bit more challenging to troubleshoot without spare parts. If you happen to have another button laying around that's compatible with your iPad, you could try that and see if that registers clicks. If it does, you probably have an issue with your original home button. However, it's probably unlikely that you have a spare home button that just happens to fit. I would recommend to try the things stated above first. If you still think that the actual button is the issue, I'm afraid taking it to Apple would be the only option if you want to keep Touch ID.*

*Each home button is paired with each iPad, meaning that replacing the original button will result in Touch ID not working. If that's your case, then sadly there's not much you can do yourself. The only way to fix that would be to do the repair through Apple. But just a heads up, according to their repair policy, they don't always accept to repair a device, no matter how much you pay them, if it has been opened up by someone that isn't an Apple Authorized Service Provider.

Good luck in fixing your device!