I think you've had the basic answer that wifi that comes from a hub/router connected to your phone line won't work in a power cut as the hub needs electricity too.
Someone's mentioned using a mobile phone as a modem to connect to the internet. That would depend how wide an area is affected by the power outage. The mobile phone masts need electricity too, and if the only ones within range of your 'phone are off, then you can forget that too.
During the storms that hit northern England last winter, Storm Desmond knocked out power where I am for about 4 days (though it felt much longer). This took out all the mobile phone masts too. The only link to the outside world (apart from driving to an area with power - though as there were impassable floods on the main roads out that wasn't much of an option) was a wind up radio, the landline telephone (once I'd unplugged my modern cordless 'phone and plugged in an old-fashioned one that got all the power it needed from the telephone network) and the car radio.
After a while the power company were able to set up a number of massive generators, though predictably my house wasn't covered by any of them, but I could as a BT Broadband customer wander the streets where there was power, looking for BT-WIFI access points.
Otherwise I read a few books, used up a lot of candles and simply went to bed when it got dark.