System pager isn't showing for logged in users
Finally figured out the problem with some help from a friend. It was being caused by a conflicting pager in a custom made View that rendered the current user's picture in a block on every page. It's only set to display when the user is logged in, hence why the problem was non-existent for anonymous users.

Changing the pager settings for the view from "Full, display 10 items" to "Display all items" has fixed the problem (presumably since there's only a maximum of 1 user picture to output in the View, Views dictated that there was only 1 page and that interfered with the other pager).

I suppose another solution, although not necessary in this case, would be to change the View's Pager ID.

In hindsight the solution is pretty obvious, because I have known about the Pager ID setting for a long time, and even knew what it did - but hopefully this will come in useful for anyone else with a complex site layout running into the same problem. :)