ILS (LOC + G/S or any combination thereof) is a selected mode. It's fully selected as in, it is not a managed mode.
It is completely dependent on ground based navaids, not the FMGS/GPS inputs.
If you've multiple Nav downgrades/electrical issues etc., you'll still have an ILS on 1 available. You'll fly to the LOC in HDG mode (under vectors or whatever) and it'll then follow the completely independent signal offered up from the airfield.
Think, say your aircraft is in perfect shape, it's locked on the LOC and GS and then the signal is interfered with etc on the ground. What'll the plane do? It'll follow the signal even though all GPS/FM position inputs are correct. The aircraft is not under the guidance/control of the flight management and guidance computers.
Likewise, if you're shooting an RNAV approach to a runway with an operating ILS signal and the ILS signal becomes erroneous, what'll the plane do? Continue tracking the RNAV correctly because it's using the GPS/FM inputs.