I suppose it's a testament to the thoroughness of the unofficial patch that this is apparently my biggest problem with the game, but this is something that's been bugging me for a while since I started playing again. The character started out at level 20, and I know multiple in-game years have elapsed, but it's still stuck at 20.

Is it like this in every game, i.e is the age just a static thing that never changes? Or is there some way to get around it?

There's no way to get it to change, and I always thought this was why the NPC ages were disabled originally, as they never change either. PC's age is always 20 + race modifier.