There is no need to do anything with the phone yet if they don't want to, they can just cancel the broadband service that is running on it currently and keep paying line rental. Cityfibre won't turn that service off unless you've told the new ISP to port the number, in which case the answer to how to connect it all up is "it depends". The BT ringers do work with ATAs though.
That would mean paying for both services, and the OP requested cheaply.
Depending on which ISP they use, their new provider might or might not have "native" VoIP support, and any new router supplied might or might not have a 431A socket for the phone. If the router has a socket it might well be locked to the ISP's digital voice. If the router doesn't have a phone socket, use an ATA to connect the phones. But installing FTTP and ceasing a copper line is the ideal opportunity to port the phone number to a 3rd party VoIP provider, which will almost certainly be cheaper than an ISP offering if they shop around, and decouples phone and broadband suppliers so that switching one or other in the future is simpler.