"It is absolutely true that Apple will replace the iPhone 6 screen for $109--and that this includes a new home button/touch id sensor and the reprogramming of the phone for the new touch ID.
This is your only option to regain touch function if you have damaged your original touch ID flex.
Read more about this coverage here, and no it doesn't require Applecare +
OK so to answer your question directly, the touch ID not working correctly after LCD replacement. The problem is not in the tough sensor, the problem is in the LCD you replaced it with NOT being an OEM replacement. There are OEM parts out there. OEM stands for parts made to the ORIGINAL EQUIPMENT MANUFACTURER specifications. If the part is not OEM then it may not work with the newest versions of iOS. What you will need is an OEM LCD replacement. If you did purchase an OEM replacement then you may have just gotten a bad LCD. I also want to stress that purchasing an OEM screen does not mean purchasing it from Apple. There are plenty of suppliers selling legitimate parts.