The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products.
Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\delta_n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$
$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$ $$\cong \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$
We also need $\otimes$ to commute with coends.
Now in the diagrams in the question the monoidality maps will have the opposite direction since $|-|$ are colax, not lax. If $\delta_{F\Delta_n}$ equals to $\delta_{\Delta_n}$ up to the isomorphism $F(\Delta_n) \cong \Delta_n$, then the first diagram will commute because (writing $.$ for the tensor)
$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$ $$=$$ $$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta).$$ The second diagram commutes because of a similar argument.
If the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and $\tau$ will become a monoidal transformation. In particular this is the case in the simplicial-topological example.
For the purpose of defining a lax structure on $|-|$ we could again look at the special cases. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps
$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$
which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via
$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$ $$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$
However, I can't tell how these can be generalized to a monoidal context, or how they can be related to each other.