At risk of spoiling a 42 year old story, I have to wonder if they've been reading too much William Gibson; Burning Chrome in particular.

Each satellite transmits its location (ephemeris) as well as the exact time in each 30 second message. This means that the GPS device does not need to calculate the location. Though the ephemeris updates every 2 hours, a value is considered valid for 4 hours.
The Almanac, a list of the status and rough location for each satellite, is also part of the message and takes over 12 minutes for complete transmission (25 messages). It is used to help determine which satellites to look for on acquisition amongst other things, though this is not essential with modern equipment.
This means that the date is not used to adjust the position information as that is part of the message itself. The alert is nothing to do with general GPS stopping working, but rather anything that depends upon the GPS signal to get the UTC time being at risk (an operation requiring the Almanac).

The information is from the Navigation Message section of the Wikipedia article on GPS Signals (https://en.wikipedia.org/wiki/GPS_signals#Navigation_message), which is very detailed.

But will it run on a NAVI?

An interesting NHK World documentary about Japanese calculator culture and the history of calculators in Japan. I generally watch these at speed = 1.5.

Begin Japanology (13 June 2013) - Calculators