Data is either paired or not. Consequently, it is either appropriate to use the paired test$-$Wilcoxon signed-rank W test, or the unpaired test$-$Mann-Whitney U test (A.K.A. Wilcoxon rank sum test), but never both. It is possible to ignore pairing and erroneously use the less powerful rank sum test. That has the tendency of spuriously increasing the probability of significant results and decreasing the probability of highly insignificant results, and although that might have meaning in some other context, e.g., contrasting Wilcoxon with t-testing for example, it has no meaning here.
It is difficult to imagine a common scenario in which one would mistakenly use the paired test on unpaired data as typically the groups contrasted have unequal sizes. Frankly I do not know what would happen if the sizes were equal and one mistakenly used the paired test. It is somewhat pointless to ask as well, as one should not be doing that.
If one wants to see effect size despite having a motive for NOT doing so, i.e., it is an error, one might think that one could do a Monte Carlo simulation to explore it. My personal problem in doing so is that if I start with a false premise, the result of such a simulation would not be logical, which makes it conceptually difficult for me to evolve appropriate other assumptions to use to set up that simulation. Perhaps that is only my personal problem, so I ask, and what exactly does one assume to set up such a test? For example, randomly correlated pairing? If so, randomly correlated how? Should the correlation itself be modeled as random, or should it be random correlation that arises from random data?