How to speed up Fallout 4 loading time
It's a well known issue related to poor coding.
The solution is to install a mod, such as great acceleration of loading times or load accelerator.
What's happening is that loading is somehow tied to your framerate, and the framerate is limited to 60 FPS. You could unlock the framerate to fix this issue, but the game's physics engine does very poorly with framerates that are too high.
The mod in question therefore unlocks your framerate during loading screens, and resets it to 60 FPS after loading has concluded. Lowering your graphics settings may further improve your loading times. Fortunately, there's not much happening on the loading screen, so lowering your graphics may not be necessary in most cases.
On my PC, the Commonwealth used to take over a minute to load. With the mod installed and working (check your FPS during loading screens), loading times were reduced to only tens of seconds (~200 - 300 FPS).
I have noticed this as well, and I have (more or less) proof: Try to uncap frame rate and enter a new area. The loading time will be significantly lower. As OP says, when capped to 60fps, the loading screen (and its animation) will be much slower and wait so you have time to read the tips. Uncapped, it's almost impossible to read.
The reason for this is because the game speed is related directly to frame rate in this engine, which means going 120fps will double effective game speed. Since the loading screen itself is a very simple animation, it can reach 240+ fps easily, resulting in lower loading times.
Of course, the the same is also true for regular gameplay: NOT capping the game to 60fps will sometimes suddenly make the game speed up and make certain mini-games (like the lockpicking one) a lot more difficult. It's a shame we can't have both, but it seems we're stuck with either unnecessarily long loading times, or sped-up gameplay.
Hopefully, someone will mod this out as soon as possible.