Hello Starlet
I cannot guarantee that this also goes for the classics, but this is the 737-800:
The ILS receivers are coupled to DC bus 1- and 2.
If you look at the electrical schematics you can see that the DC Bus 1- and 2 are separated by a cross bus tie relay. When you arm 'appr' and capture the glideslope this cbtr opens to ensure that failure of one electrical source doesn't affect both ILS receivers.
With one engine out and APU inop, the ILS receiver on the side of the failed engine receives power due to the closed cbtr.
Upon glideslope capture this cbtr opens and isolates the two ILS receivers, leaving one without power.