VOIP has exactly the same problem. I thought it was a requirement to have battery backup on PSTN lines?!?!?
Did you mean that?
PSTN lines are the copper ones, with power provided by the exchange.
The requirement is, I think, actually written in terms of providing a robust enough service for access to the emergency services - but last time I looked, it doesn't actually specify what "robust enough" would qualify as.
When voice is provided within BT's FVA service (voice over fibre), I think their interpretation of the requirement is that it requires a backup battery with a few hours of life. This is a form of VoIP, but is carried within a separate VLAN in the access network and at the exchange, the voice component is then carried separately within the (more robust) core network.
However, your average VoIP service carried "on top of" your average internet connection does not have this robustness. It is as reliable as the "best effort" nature of the internet as a whole.
If your ISP fails (eg you can't authenticate), then FVA would work, but VoIP carried over the internet connection would fail.