There are several questions above.  The answer below gives one multigraded finite free resolution of multigraded finitely generated modules over the multigraded polynomial ring that are <I>saturated</I>, i.e., equal to the multigraded module of a coherent sheaf on the associated product of projective schemes.  Since the Hilbert polynomial of a finitely generated multigraded module equals the Hilbert polynomial of its saturation, this suffices for the application to multigraded Hilbert polynomials.

One useful result is Beilinson's <I>resolution of the diagonal</I>.  Since this is completely functorial, it helps with these types of questions.  Let $V$ be a $k$-vector space of finite dimension $n$.  Let $r$ be an integer with $0<r<n$. Denote by $$(G\to \text{Spec}\ k,\ q^\dagger:V^\vee\otimes_k \mathcal{O}_G \twoheadrightarrow S^\vee)$$ a universal pair of a $k$-scheme together with a surjective homomorphism of locally free sheaves whose target has rank $r$.  The adjoint homomorphism fits into a short exact sequence of locally free sheaves, $$0 \to S \xrightarrow{q} V\otimes_k \mathcal{O}_G \xrightarrow{p} Q \to 0.$$ On the self-product in the category of $k$-schemes, $G\times G$, there is an associated homomorphism of locally free sheaves, $$\text{pr}_1^*p\circ \text{pr}_2^*q:\text{pr}_2^*S \to V\otimes_k \mathcal{O}_{G\times G} \to \text{pr}_1^* Q.$$ By adjointness of Hom and tensor product, this is equivalent to a homomorphism of locally free sheaves, $$\alpha:\mathcal{E}
 \to \mathcal{O}_{G\times G}, \ \ \mathcal{E}:= \text{pr}_1^*Q^\vee \otimes_{\mathcal{O}_{G\times G}} \text{pr}_2^*S.$$ Beilinson's observation is that the image of $\alpha$ is precisely the ideal sheaf of the diagonal in $G\times G$, which is itself everywhere locally cut out by a regular sequence of length $m=\text{dim}\ G = \text{rank}\ \mathcal{E}=r(n-r)$.  Thus, the Koszul complex of $\alpha$ gives a finite, locally free resolution of the structure sheaf of the diagonal, $$\left( K_\ell(\alpha) := \bigwedge^\ell_{\mathcal{O}_{G\times G}}\mathcal{E} \right)_{0\leq \ell \leq m}, \ \ (d_\ell:K_\ell(\alpha) \to K_{\ell-1}(\alpha))_{1\leq \ell \leq m}. $$
Note that the exterior power $\bigwedge^\ell \ \mathcal{E}$ has a direct sum decomposition as a tensor power of Schur functors $\text{pr}_1^*\mathbb{S}_{\lambda}(Q^\vee)\otimes_{\mathcal{O}_{G\times G}}\text{pr}_2^*\mathbb{S}_{\lambda'}(S),$ where $\lambda$ runs over all partitions of $\ell$ into integers no greater than $r$ and with no more than $n-r$ parts.

The same holds for any product of projective spaces, i.e., for an integer $\rho \geq 1$ and an ordered $\rho$-tuple of ordered pairs $(V_i,r_i)$ as above, for the product scheme $X=G_1\times \dots \times G_\rho$ with its coordinate projections $\pi_i:X\to G_i$, on the product $k$-scheme $X\times X$, one locally free resolution of the structure sheaf of the diagonal is the Koszul complex of the homomorphism of locally free sheaves, $$\beta: \mathcal{F} \to \mathcal{O}_{X\times X}, \ \ \mathcal{F}:= \bigoplus_{i=1}^\rho (\pi_i\circ \text{pr}_1,\pi_i\circ \text{pr}_2)^*\mathcal{E}_i ,$$
whose components are the maps $(\pi_i\circ \text{pr}_1,\pi_i\circ \text{pr}_2)^*\alpha_i$.  Note that each locally free sheaf $\bigwedge^\ell \mathcal{F}$ has a canonical direct sum decomposition into tensor products of pullbacks of locally free sheaves $\bigwedge^{\ell_i} \mathcal{E}_i$, indexed by all ordered $\rho$-tuples of nonnegative integers $(\ell_1,\dots,\ell_\rho)$, such that $\ell_1+\dots+\ell_\rho$ equals $\ell$ and each $\ell_i\leq m_i$.  
One application of this is the following.

<B>Theorem [Beilinson resolution].</B>  Let $\mathcal{G}$ be a coherent sheaf on $X$ such that for every ordered $\rho$-tuple $(\ell_1,\dots,\ell_\rho)$ as above, all higher cohomology vanishes for the tensor product of $\mathcal{G}$ and the pullbacks $\pi_i^*\mathbb{S}_{\lambda_i}(Q^\vee_i)$, where each $\lambda_i$ is a partition of $\ell_i$ into integers no greater than $r_i$ with at most $n_i-r_i$ parts.  Then the complex $\text{pr}_2^*(K_\bullet(\beta)\otimes_{\mathcal{O}_{X\times X}}\text{pr}_1^*\mathcal{G})$ is a locally free resolution of $\mathcal{G}$ whose terms are direct sums of tensor products of locally free sheaves $\mathbb{S}_{\lambda'_i}(S_i)$.  This resolution is functorial in $\mathcal{G}$. 

A particularly important case is when each $G_i$ is just a projective space, i.e., every $r_i$ equals $1$.  In this case, the terms of the locally free resolution above are direct sums of tensor products of invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(-d_i)$ for $0\leq d_i < n_i$.  Since the tensor product of $\mathcal{G}$ with the invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(e_i)$ satisfies the hypothesis above for all integers $e_i \gg 0$, this gives the following corollary.

<B>Corollary.</B>  On a product of projective spaces, $X=\mathbb{P} V_1 \times \dots \mathbb{P} V_\rho$, for every coherent sheaf $\mathcal{G}$, for all ordered $\rho$-tuple of integers $(e_1,\dots,e_\rho)$ with $e_i\gg 0$, there is a functorial locally free resolution of the tensor product of $\mathcal{G}$ and the invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(e_i)$ as above.  After tensoring this locally free resolution with the tensor product of the invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(-e_i)$, this gives a functorial locally free resolution of $\mathcal{G}$ whose terms are direct sums of tensor products of invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(-d_i-e_i)$ with $0\leq d_i < n_i$.

Translating this back into the language of multigraded modules over the multigraded homogeneous coordinate ring of $X$, this gives a functorial finite free resolution by direct sums of shifts of the multigraded homogeneous coordinate ring.

As mentioned in my comments above, one needs <I>much less</I> if one only needs to prove the existence of a multigraded analogue of the Hilbert polynomial.  However, the functoriality in the Beilinson resolution is frequently useful.