Like all electronic devices, wireless access points need power to function, so a WiFi hot spot will not work on its own during a power outage. Plugging the device into an uninterruptible power supply will maintain its function throughout an outage.
As long as the access point has backup power, all devices connected to it will be able to communicate normally in the event of a power outage. Any other devices required for Internet connectivity, such as cable modems, routers or desktop computers, also need backup power supplied.
If the outage is caused by a disaster that severs communication lines, then Internet connectivity will be unavailable even if devices are powered locally. Devices such as cell phones and LTE hot spots that use cellular towers to provide an Internet connection will function normally during an outage as long as their batteries allow.