Does Coordinate time make sense in a universe with wormholes?

Let us say that you have a spacetime with wormholes, and a coordinate system. The two ends are created at almost the same point in spacetime. Then the one end is taken far away and time dilated; (note: this is all in reference to the coordinate system). How would coordinate time "work" in this case?

The reason I ask is that there appears be a contradiction in applying coordinate time. Consider the paths of the wormhole ends. If the end that didn’t move aged a time of $t$, then the other end would age a time of $t’ < t$, due to the time dilation. This is the premise of the wormhole time-machine: by time dilating the one end, you have a way to travel to a different time (according to coordinate time). However, it is not clear how coordinate time works, since the ends of the wormhole touch each other in spacetime (that is what a wormhole is). Is the inside of the wormhole coordinate time $t$ or $t’$?

How do you go about defining coordinate time in this situation?

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My guess is that you can still define a differential $dt$ though of as the differential of coordinate time, but that this does not consistently define a coordinate time globally, since you can have spacetime loops with $Delta t = int {d t}$ that are non-zero. Or maybe each point of spacetime has more than one coordinate and therefore more than one coordinate time?