Not a stupid question at all - after all St Catherine's Hospital is not listed, despite being a workhouse originally. Must be too many examples of that style from that era.
I did a bit of work in this building when I was an apprentice about 15 years ago. It was not a very pleasant place. It was quite dirty and most of the space on each floor was empty by that time. There used to be a bizarre lift which consisted of little more than a vertical conveyor belt running up through all the floors with handles and steps on it. The belt ran continuously and you literally had to jump on and then off at your floor. The Health and safely brigade these days would have a ceasure if they saw it. The view from the roof was great though.
I had to repair some machines at Bibby's (?) Mills in Liverpool moons ago. They had the lifts you describe. They were called Paternosters. Frightening to use the first time, especially lugging a heavy toolbox in and out. You had to get your timing just right !
If you bung "Paternoster, a Cyclic Elevator" into Youtube, you will see they are still in use.
I'm amazed the H&S haven't pounced - yet. Thanks Kev for awaking an old memory!