Cramér's $V$ doesn't correspond well to what is tested by McNemar's test.
Edit: Disclosure: the webpage and R package cited below are mine.
Probably the most common effect size statistic for McNemar's test is odds ratio, though Cohen's $g$ could be used. Cohen (1988) also uses a statistic he calls $P$.
For defintions, quoting from here:
Considering a 2 x 2 table, with $a$ and $d$ being the concordant cells and
$b$ and $c$ being the discordant cells, the odds ratio is simply the greater of $(b/c)$ or $(c/b)$, and $P$ is the greater of $(b/(b+c))$ or $(c/(b+c))$. Cohen’s $g$ is $P – 0.5$.
Cohen (1988) also gives interpretations ("small", "medium", "large") for his $g$ statistic. Because $g$ is monotonically related to odds ratio, these interpretations can be extended to the odds ratio (same link). Edit: Interpretations of effect size statistics are always relative to the field of study, specific experiment, and practical considerations. Cohen's interpretations should not be considered universal.
Obviously, not much coding is needed to calculate any of these statistics for the 2 x 2 case. However, if they are extended to larger tables, the math can get tricky. In R, the
cohenG() function in the
rcompanion package makes this relatively easy.
Links include R code.
Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences, 2nd Edition. Routledge.