because LOC and GS work the same way, if you were to fly one at 90 degrees bank would your information be displayed on the wrong indicators?
No, the directional part of ILS is located at the ground. (Ground) transmitter creates lobes with specified frequency and modulation to mark the high, low, right and left areas.
LOC and GS signals are distinguished by the carrier frequency, deviation from the optimal approach by relative intensity of 90 Hz and 150 Hz modulation in the signal. These signals provides information about position relative to the airport regardless of airplane attitude.
So flying knife-edge LOC axis of the instrument still shows (correct) horizontal deviation and G/S axis correct vertical deviation.
Of course, display of the instrument is turned 90° together with the whole airplane, so its G/S axis now lines up with the green-blue boundary you see behind your windows (I hope you are not flying such maneuvers in IFR conditions :) ) but it is still the G/S axis and it shows vertical deviation from the optimal glideslope.
Only difference would be that airplane antenna won't match the transmitter polarization (ILS transmitters transmit in horizontal polarization and receiver is likely polarized in the same plane), so you loose some signal intensity and receiver can be more sensitive to reflected signals, therefore reducing maximal working range and possibly increasing probability of spurious signal detection.
As long as the aircraft accurately flies down the ILS, it doesn’t matter if it is knife edge, inverted, etc. The indications on the GS and LOC will stay centered. If you deviate from the GS or LOC the deviations will be shown on the respective instruments. The instruments don’t know or don’t care what the aircraft attitude is. The instruments only care about the deviation.
LOC and GS work the same way
They work on the same principle, but they each have their own radio signal. The localizer is on the frequency tuned in the radio (in range 108.10 MHz to 111.95 MHz), the glide-slope is on an associated frequency from the 329.15 MHz to 335.00 MHz (the mapping is not sequential for some reason). So the localizer needle will still show the deviation from the localizer signal and the glideslope needle will show the glideslope signal. They just won't be aligned with the directions to steer any more.
Neither also knows anything about direction. Each carrier has two tones broadcast on it (90 Hz and 150 Hz) that indicate left and right on the localizer frequency and high and low on the glideslope frequency. So when you are to the left of centreline, the receiver will sense more 90 Hz on the localizer and the localizer needle will move to the right relative to dashboard, no matter how your plane is oriented.
Also due to the nature of phase-shift directional antennas, there are other maxima around besides the main ones just around the intended flight path. One especially has to be careful about the false glide-slopes at 6° and 9°. The 6° has reverse sensing, that is above it the needle will direct you to the 9° one, below to the correct 3° one. And 9° is way too steep for any aircraft. There are similar maxima for the localizer, called side-lobes, but they are a bit easier to avoid by comparing your heading with the runway number.
To try to make things totally clear:
Suppose that the aircraft is making an approach in knife-edge flight, banked 90 degrees to the right. Suppose that the runway heading is north (it's runway 36).
If the aircraft is too low, the glideslope needle in the course deviation indicator will deflect "up" (towards the ceiling), indicating that the aircraft is too low. The pilot will correct for this by adding more left rudder input (which is to say, pressing harder on the left rudder pedal).
If the aircraft is too far to the east (right), the localizer needle in the CDI will deflect "left" (towards the left wing), indicating that the aircraft is too far to the right. The pilot will correct for this by giving down elevator input (which is to say, pushing forward on the stick).
The readings on the CDI will be correct in that they will correctly describe the location of the aircraft relative to the glidepath. However, they will not directly indicate what the pilot has to do; the pilot will have to recognize that, since they are flying in an unusual attitude, they will have to react to the CDI indications in an unusual way.