Within the constraints of general relativity this is a solvable problem, albeit a stupidly hard one. – Joe Bloggs
I agree with this assessment.
Wormholes are theoretically consistent with general relativity, but they are speculative.
So even if one were to calculate all the details of the proposed scenario there is no data to base this on.
If you have ever seen any calculations of relativity, you know that it quickly becomes complicated if you want to do more than calculate time dilation for two differently moving bodies.
The proposed scenario would definitely be a very complicated one and even if anyone (definitely not me) would post it here in an answer there would only be a very low number of readers able to understand it.
Wormholes connect two points in spacetime, which means that they would in principle allow travel in time, as well as in space. - Wikipedia
Note that they are not "points" as in singularities, but they stretch in all 4 dimensions. This means they occupy a 3 dimensional space over a span of time. (Relativity makes it a little more complex, but let's keep it at that.)
Coupled Wormholes would remain a constant link between two of these 4 dimensional portions of space-time.
While I personally have no idea how to represent that in math, I see no direct problem in why a smaller portion of space-time couldn't 'move' through another as long as neither changes state in some way that it would break the link.
I don't see a problem, since - if we already have traversable wormholes - objects and other forms of energy can move through it, why would another wormhole be different? I believe the scientists who have shown that theoretically objects can move through unharmed and without inconsistency with current knowledge, so I see no reason why a space-time link would be any different.
I think so because for any given moment you can define where the wormhole is, even if it just partially traversed the other wormhole and if an object would move through the inner wormhole in that moment it would not be changed at all, even if it is only half through the outer one - because we already assumed it could move through the other without problems.
However the mere concept of this is two layers of hypothesis and no hard data to back any of this up.
I do not think there is an apparent inconsistency with the scientific agreement that wormholes are theoretically possible and if you use that within a world you created you wouldn't create logical inconsistencies.
A little bit of hand-waving is probably required anyway, unless you want to do the math of relativistically consistent wormholes for yourself and possibly readers.