I asked the question of internet staying up during power cuts when I spoke to CityFibre recently. They "opined" that IF I could power up my modem/router and my VOIP box then I SHOULD be able to make/receive phone calls over the internet as normal. The only shame is that our latest router is powered from a USB phone charger at 5 volts and not the 12 volts that the previous ones have been. 12 volts would have been so much easier to manage with a left over car battery and a trickle charger set up. A sort of DIY UPS. I am still tempted to make a 5 volt supply with battery backup to support the modem/router and wait for a power cut to afflict us.
Rans6................