Wolfox

I would assume that it would just see it the same way anyone other than Jotaro/DIO see it in real time: instant skip between the activation and the end of the stopped time

Not fully. As seen in the Anime Giorno occasionally noticed the inconsistencies. Granted not enough to be a huge downside, but it is somewhat noticeable 

That's for king crimson though, where the skip is noticeable to anyone affected because of the lack of perception or the erased time? With The World being a pause, it's only noticeable from the actions of the one moving in stopped time. 

One interesting thing I've thought about a couple times from this is that it shows that King Crimson is limited range, while The World is unlimited.
If The World was limited range, there would be noticeable shearing at the boundaries where everyone just outside would see everything inside the range freeze. With a universal pause, there's no way to perceive it outside of seeing the effects of the users actions, which matches what we've seen. For King Crimson, it's the reverse. We've seen that anyone affected by King Crimson is able to at least somewhat notice its activation (because they have no memory of the last 10 seconds) even if completely out of range of Diavolo himself. and he's used it plenty in order to reach the position he's in, so if that was affecting the whole world every time he used it, people would super know about it, which is clearly not the case.