There are three categories of ILS. The most basic one can get you down to 200' AGL and can be flown using only the signals from the localizer and glideslope antennas. These are the 110 and 330 MHz systems, respectively. They are transmitters sited at the airport, and there is receiving equipment in the aircraft. The aircraft uses a standard barometric altimeter. (Technically a "sensitive" altimeter but this distinction is pretty much irrelevant if you're flying anything made after 1930.) No radar altimetry or other special equipment is required; small single-engine planes flown by appropriately rated private pilots fly these all the time. These systems aren't affected by 5G.
The other two categories get you down to 100' AGL and 0' AGL, and these are the ones at issue. These require additional equipment, training, operations manuals, and certification of air crew. They also use the localizer and glideslope, but they require a radar altimeter as well (plus a bunch of other additional equipment in the aircraft, mostly for redundancy, that isn't relevant to this discussion). The radar altimeter is C Band and has no ground components.
The 0' AGL category is full autoland, with the autopilots controlling the aircraft not just all the way down to the ground, but throughout the landing rollout too. I suspect these are the "fly by wire" systems people are referring to, which is a bit of a misnomer if that's the case. "Fly by wire" is when the stick/yoke has no mechanical-hydraulic connection to the flight control surfaces; instead it provides signals to a set of flight computers, which mix those inputs with signals from the aircraft's sensors to produce an output for the actuators. It's possible that the radar altimeters are included in the inputs to the flight computers but I don't know. (I'd be surprised.) Autopilots, on the other hand, are more properly referred to as "automation." These would definitely be affected by problems with the radar altimeters during Cat II & III ILS approaches.
Edit: Some Cat I (basic) ILS approaches are marked "SA" meaning "Special Authorization." These approaches do require radar altimetry, but they're the exception when it comes to Cat I.