As @B.fox explains in his comment, our understanding of wormholes does not include another dimension. When you pass through a wormhole, you are still in 3D linear time space.1 However, it is thought that gravity must still exist in a wormhole. Therefore, your question may still be answerable.
I suspect people think that wormholes pass through other dimensions thanks to movies like Interstellar, which correctly portray the opening of a wormhole as a sphere (3D), not a hole (2D, think Star Trek: Deep Space Nine, ), in space ... This might lead people to conclude that traveling through it must push you to a "fourth" or another dimension. This is not the case, insofar as we understand wormholes. An amazingly simple presentation of wormholes is found at Space.com
Think of a wormhole as a shortcut. Space is twisted around, like twisting a piece of paper, to bring two points closer together. Because the effects of gravity still exist in a wormhole, if that shortcut "passed near" a black hole, your passage through a wormhole would experience time dilation.2 If (much more likely) one of the mouths of the wormhole were near a black hole, then, too, you would experience dilation. No matter where the wormhole forms, if you pass through it with enough velocity, you will experience time dilation (this assumes the structure of the wormhole will allow for such speeds...).
As for whether or not time dilation can happen at higher dimensions... That depends on what you mean by a "high dimension" (and since we have yet to empirically prove they exist, it's pure speculation). One way of looking at a "higher dimension" is to add dimensions of existence, e.g., time. A building is 3D, but its existence is 4D due to the passage of time (it begins new and changes or decays during a measurable period of time). Because this perspective is no different than the lives we live now, time dilation is the act of modifying the 4th dimension, which isn't something you "pass-through" as much as it is something you "measure."
If what your talking about is a dimension of physical expression, such that a tesseract is a 3D shadow of a 4D cube having four lines drawn from the same vertex, each at 90° offset from the other, then we really don't know other than to conclude that such a cube must still have mass, and therefore gravity, and therefore could be affected by the gravity of a black hole, which means travel through that dimension would still experience time dilation due to proximity to a black hole or speed.
Conclusion
Yes, you can experience time dilation when traveling through a wormhole and yes, if you consider other dimensions to be dimensions of physical expression, then you can experience time dilation in those dimensions, too.
Finally, let me offer some advice. A good story has more to do with how you write it than it does the precision of the science you use as its foundation. A good example is Star Trek IV: The Voyage Home, where Checkov and Uhura collect photons through a closed bulkhead door. We, the moviegoers, don't really care that what they're doing makes no sense because we're enjoying the story, not the science. Even in Science Fiction, science is only the window dressing that lets you tell a good story. Don't get too hung up on whether theoretical science supports the story you want to tell. Focus on telling a good story.
1 A lot of assumptions are occurring here. The most common example of demonstrating a wormhole is to fold a piece of paper, basically creating a 3D shortcut through a 2D existence. Maybe this means when space is folded we're taking a 4D shortcut through a 3D universe ... maybe not. I'm fond of a scene from Stargate SG1
where aliens are talking about a wormhole with Samantha Carter. "Oh," she exclaims," you mean like an Einstein-Rosen bridge!" "No," responds the alien, "it's not like that." Underscoring the simple reality that we barely understand what we're talking about and claiming wormholes do or do not take us through another dimension is jumping to a conclusion.
2 The problem with this concept is that if space is only "folded" like folding a piece of paper, then it can't happen. If space, being 3D, is "twisted," then it might. It's a murky subject, and explaining it as if space were 2D is very much part of the problem.
