How does an FMS shift after losing all external navaids?
Turns out indeed it's an issue affecting polar operations. It differs from type to type, but for the 737-600/-700/-800/-900 as an example, the message UNABLE REQD NAV PERF-RNP will be displayed.

For oceanic areas this will only happen if ANP shifts by over 10 NM, i.e., after losing the RNP-10 capability.

At an IRS drift rate of 650 metres per hour, it would take 28 hours of being airborne or a rather shifty IRS unit—no pun intended—for the message to be displayed.

So the answer is that the magenta line won't shift, unless the pilots choose an IRU over the rest by using the position shift function of the FMS.

For the 747-400:


  When no global positioning system (GPS) updating occurs, all position and velocity corrections gradually are phased out until the FMC navigation parameters equal the selected IRU position and velocity.


This logic is part of the RNP standards and regulations, to stop an auto flight system from chasing the course when navigational accuracy becomes questionable. Any loss of GPS (away from ground navaids) inhibits the FMS from applying any corrections without crew intervention.



Source: Polar Route Navigation