I am jumping in the fray. I think what Dog is saying is that where really isn't a relationship in an official sense. Truly WT facilities are the sport element of TKD and has it's own body of regulation. The Kukkiwon, as part of the Korean government, has it's own body of control. Because both have very strong Korean roots and are both often taught together at the same schools, they get squeezed together as the same entity. As an example, as black belt certificate doesn't come from WT, it comes from Kukkiwon.
Officially, no. But from the functional perspective there is.
I work IT at a hospital, and there is the "technical" and "functional" side of things. Technical is things like "can the user log in" and "do the servers talk to each other." Functional is things like "does the layout and terminology make sense to the doctors" and "are these workflows optimized?"
So while technically the organizations may be separate, for the student, they train in TKD forms and sparring, and they go to tournaments and compete in TKD forms and sparring. The tournaments register black belts based on their KKW ID, and we teach "Olympic-style sparring" at our school.
There's a system at my hospital, which other systems can connect into it. (Think of it like how you use Facebook, and in addition to the main Facebook page and apps, there's also 3rd-party apps and games you can use, like Words With Friends). Facebook doesn't make Words With Friends, and technically it's a 3rd-party app that's contracted through Facebook. However, because you use your Facebook account and you can play it on the Facebook site, most of the players see it as a Facebook game. Similarly, the system that I manage is completely separate from the system it's used in, but because the accounts are linked and you use the same application to access both of them, the users see them as the same thing. (And in my case, the developers of both systems have so little to do with each other they won't even talk to each other, I have to be a middle-man when I escalate anything).
With that said, when I look at it from the perspective of a TKD student, or even an instructor or coach, whether the two organizations are so separate they merely read the news about each other, or if they're run by the same person and are two branches of the same affiliation is irrelevant. The fact is, how it works out for us in practice, is what we train at a KKW school is what we compete in at a WT tournament, and our qualifications at a KKW school is what we use to sign up for the WT tournament. So even if they're technically separate, that doesn't matter to the end user...er, the student/competitor.
From the top down, you might see the divide. But from the bottom up, you see a Venn Diagram with a very large center section, if you even realize there's a difference.