All landlines are changing to VOIP (voice over internet protocol) and the old telephone lines will be redundant. To continue using your phone you will need a VOIP adapter that connects your phone to the router. I don't know if BT will provide that or if you will have to buy one.
I also don't know when your telephone line will be disconnected but the fact that BT want to put fibre in your house and is supplying new handsets indicates that the change is imminent. You really do need to talk to BT and find out exactly what is going on.