On the one hand, the proof is very cheap. Let $Z_j=e^{2\pi iX_j}$. $X=\sum_j X_j$, $Z=e^{2\pi i X}$. Note that $\operatorname{Var}_{\mathbb R/\mathbb Z}X\approx 1-|EZ|$ and similarly for $X_j$ and $Z_j$. Now just use the identity $EZ=\prod_j EZ_j$ to conclude.

On the other hand, finding the reference may be a highly non-trivial task, so I leave it to somebody else :-).

On the one hand, the proof is very cheap. Let $Z_j=e^{2\pi iX_j}$. $X=\sum_j X_j$, $Z=e^{2\pi i X}$. Note that $\operatorname{Var}_{\mathbb R/\mathbb Z}X\approx 1-|EZ|$ and similarly for $X_j$ and $Z_j$. Now just use the identity $EZ=\prod_j EZ_j$ to conclude.

On the other hand, finding the reference may be a highly non-trivial task, so I leave it to somebody else :-).

P.S. What can be really concluded here is that there exist $\gamma_j$ summing to $\gamma$ with $\sum_jE\|X_j-\gamma_j\|_{\mathbb R/\mathbb Z}^2\le C\varepsilon$. The condition $EX_j=0$ does not allow one to conclude from here that we can take $\gamma_j=0$: take $X_j$ symmetric with values about $\pm \frac 12$ for a counterexample.