In a word – No unless you take steps to ensure it will.
This isn’t a new situation. Many people have portable telephones in the house already and these will not work if the power is off. There is the same issue with VoIP – if there is no power to the ‘base station’ it won’t work.
If you are worried about power outages, you can purchase a small personal UPS battery backup system for your router and phone from around £45 or use a mobile phone.