ADSL is exchange based so will continue to run until the exchange UPS/Generator provision is expended.
FTTC will run until the batteries in the cabinet are exhausted.
Or the home backup, as that is what keeps the router on for FTTC.
The OP also seems to me to be referring to VOIP/Digital Voice: ("With POTS we knew the line was 'always on' for voice calls"). Therefore the phone would also rely on the home backup. As it will for FTTP/H.
Connections: OnePlus 8 Pro on Three 4+ (LTE)/5G and at home Three Mobile, with (Three)ZTE MF286D router giving about 113/20Mbps.
The best of all possible worlds?