Nexus 5 Location has not updated for months?
Okay, just figured it out. For some reason I had Location set to "Device only" and the subtext below this says "Use GPS to determine your location." Seems like this should still update the location, but like I said in my question, it simply hadn't updated the current city for months. 

SOLUTION:
Fixed it simply by switching to "Battery saving" mode. Obviously "High accuracy" would work too. 

I'm not sure why the GPS feature under "Device only" wouldn't be able to find my location, maybe my GPS component is malfunctioning. Either way, it's fixed now.