A Schwarzschild (zero angular momentum) black hole is not usually considered as related to a wormhole. Once a particle crosses the event horizon, it falls into the singularity in a finite proper time and is not expected to come out.
Wormholes are usually associated to Kerr (nonzero angular momentum) black hole. It has extra features compared to the unique event horizon of the Schwarzschild black hole, like an ergosphere, Cauchy horizon, and so on.
https://en.wikipedia.org/wiki/Kerr_metric#Kerr_black_holes_as_wormholes
Now I am not claiming to have a full understanding, but as it says in the relevant paragraph of the wiki article on black holes, if an object crosses the Cauchy horizon it might follow a trajectory that makes it exit the Kerr black hole, through another Cauchy horizon, in a region with a "normal" geometry. For observers in that region of space, it would look as if this object is emerging of a "white hole", of Kerr type, of course, not <PERSON> type.
Now this region of space is a priori not related at all with the region one has started from. It may be in a totally different universe, one to which is impossible to reach by any other way but going through the two successive Cauchy horizons described in the above paragraph if the Wiki article.
But for all we know, it is possible that this <PERSON> "white hole" is in our Universe, but it might just be anywhere, in the same galaxy, or a totally different galaxy anywhere. We can mathematically describe the path "through two Cauchy horizons" but where the exterior of the second is with respect to the first one, through "normal" space is totally out of our knowledge. 43
And quite possibly, it might well be in a "different" universe altogether.
This is what is referred to as a "wormhole" : entering a Kerr black hole here, going through a first Cauchy horizon, exiting through a second Cauchy horizon, but where ?
Maybe I should point out that a Kerr hole is not "intrinsically" black or white. It has a "past" Cauchy horizon from which an object might come out (as if from a "white hole") but also a "future" Cauchy horizon. So if we plunge into a <PERSON> "black" hole in our Universe and emerge into the "normal" space of a different one in through the "past" Cauchy horizon of a Kerr hole there, which would in this event behave as a "white hole" for external observers, one could decide afterwards to plunge again into the same Kerr hole. But it is not "going back". It would be treating this <PERSON> hole as a "black hole". We will reach its "future" Cauchy horizon, and if we reemerge in normal space, a priori it will be in a third Universe. Nothing at all guarantees we are going back ! 43