TL; DR: Suppose you have a functor between small categories $\mathcal C_0\to \mathcal C$. Let $\mathcal D$ be a locally small category such as Top or Ch. Then there is an adjunction of functor categories $$ L:[\mathcal C_0, \mathcal D]\leftrightarrows [\mathcal C, \mathcal D]:R$$ Where the right adjoint $R$ is the restriction and $L$ is the left Kan extension. There is a similar adjunction between categories of contravariant functors. Now suppose we have functors $F\colon \mathcal C_0\to \mathcal D$ and $G\colon \mathcal C^{\operatorname{op}}\to \mathcal D$. Then there is an isomorphism of coends: $$ F\otimes_{\mathcal C_0} RG\cong LF \otimes_{\mathcal C} G.$$

The isomorphism you ask about is an example of this adjunction.

------ Now with more details -------

By Elmendorf-type theorem, you can identify the category of $G_1\times G_2$-spaces that are $G_2$-free with the category of functors $[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}]$, where Top is the category of pointed spaces. Similarly, you can identify coefficient systems of the kind you consider with functors $[\Gamma_{G_1\times G_2}, \mbox{Ch}]$, where Ch denotes the category of chain complexes. More precisely, coefficient systems correspond to functors that take values in chain complexes concentrated in degree zero.

The group homomorphism $G_1\times G_2\to G_1$ induces a functor $\mathcal O_{G_1}\to \Gamma_{G_1\times G_2}$.

This functor induces an adjunction of functor categories $$[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}] \leftrightarrows [\mathcal O_{G_1}^{\operatorname{op}}, \mbox{Top}]$$ where the right adjoint is the pullback. The left adjoint, a.k.a the left Kan extension, can be identified with the functor $X\mapsto \frac{X}{1\times G_2}$. Here I have identified the functor categories with spaces with action of $G_1\times G_2$ and of $G_1$ respectively.

Similarly there is a pair of adjoint functors between categories of coefficient systems

$$[\Gamma_{G_1\times G_2}, \mbox{Ch}] \leftrightarrows [\mathcal O_{G_1}, \mbox{Ch}].$$ Here, again, the right adjoint is the pullback, and it is equivalent to the functor $F$ that you describe.

Finally, the Bredon homology groups $H_*^{G_1\times G_2}(X, FN)$ can be identified with the homology groups of the coend $$C_*\left(X^H\right)\otimes_{H\in \Gamma_{G_1\times G_2}} FN$$ and similarly the Bredon homology groups $H_*^{G_1}\left(\frac{X}{1\times G_2}, N\right)$ can be identified with the homology groups of the following coend $$C_*\left(\frac{X}{1\times G_2}^H\right)\otimes_{H\in \mathcal O_{G_1}} N.$$ The equivalence of the two coends follows from the adjunction.

TL; DR: Suppose you have a functor between small categories $\mathcal C_0\to \mathcal C$. Let $\mathcal D$ be a locally small category such as Top or Ch. Then there is an adjunction of functor categories $$ L:[\mathcal C_0, \mathcal D]\leftrightarrows [\mathcal C, \mathcal D]:R$$ Where the right adjoint $R$ is the restriction and $L$ is the left Kan extension. There is a similar adjunction between categories of contravariant functors. Now suppose we have functors $F\colon \mathcal C_0\to \mathcal D$ and $G\colon \mathcal C^{\operatorname{op}}\to \mathcal D$. Then there is an isomorphism of coends: $$ F\otimes_{\mathcal C_0} RG\cong LF \otimes_{\mathcal C} G.$$

The isomorphism you ask about is an example of this adjunction.

------ Now with more details -------

By Elmendorf-type theorem, you can identify the category of $G_1\times G_2$-spaces that are $G_2$-free with the category of functors $[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}]$, where Top is the category of pointed spaces. Similarly, you can identify coefficient systems of the kind you consider with functors $[\Gamma_{G_1\times G_2}, \mbox{Ch}]$, where Ch denotes the category of chain complexes. More precisely, coefficient systems correspond to functors that take values in chain complexes concentrated in degree zero.

The group homomorphism $G_1\times G_2\to G_1$ induces a quotient by $G_2$ functor $\Gamma_{G_1\times G_2}\to \mathcal O_{G_1}$.

This functor induces an adjunction of functor categories $$[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}] \leftrightarrows [\mathcal O_{G_1}^{\operatorname{op}}, \mbox{Top}]$$ where the right adjoint is the pullback. The left adjoint, a.k.a the left Kan extension, can be identified, once again, with the quotient by $G_2$ functor $X\mapsto \frac{X}{1\times G_2}$. Here I have identified the functor categories with spaces with action of $G_1\times G_2$ and of $G_1$ respectively.

Similarly there is a pair of adjoint functors between categories of coefficient systems

$$[\Gamma_{G_1\times G_2}, \mbox{Ch}] \leftrightarrows [\mathcal O_{G_1}, \mbox{Ch}].$$ Here, again, the right adjoint is the pullback, and it is equivalent to the functor $F$ that you describe.

Finally, the Bredon homology groups $H_*^{G_1\times G_2}(X, FN)$ can be identified with the homology groups of the coend $$C_*\left(X^H\right)\otimes_{H\in \Gamma_{G_1\times G_2}} FN$$ and similarly the Bredon homology groups $H_*^{G_1}\left(\frac{X}{1\times G_2}, N\right)$ can be identified with the homology groups of the following coend $$C_*\left(\frac{X}{1\times G_2}^H\right)\otimes_{H\in \mathcal O_{G_1}} N.$$ The equivalence of the two coends follows from the adjunction.

TL; DR: Suppose you have a functor between small categories $\mathcal C_0\to \mathcal C$. Let $\mathcal D$ be a locally small category such as Top or Ch. Then there is an adjunction of functor categories $$ L:[\mathcal C_0, \mathcal D]\leftrightarrows [\mathcal C, \mathcal D]:R$$ Where the right adjoint $R$ is the restriction and $L$ is the left Kan extension. There is a similar adjunction between categories of contravariant functors. Now suppose we have functors $F\colon \mathcal C_0\to \mathcal D$ and $G\colon \mathcal C^{\operatorname{op}}\to \mathcal D$. Then there is an isomorphism of coends: $$ F\otimes_{\mathcal C_0} RG\cong LF \otimes_{\mathcal C} G.$$

The isomorphism you ask about is an example of this adjunction.

------ Now with more details -------

You can identify the category of $G_1\times G_2$-spaces that are $G_2$-free with the category of functors $[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}]$, where Top is the category of pointed spaces. Similarly, you can identify coefficient systems of the kind you consider with functors $[\Gamma_{G_1\times G_2}, \mbox{Ch}]$, where Ch denotes the category of chain complexes. More precisely, coefficient systems correspond to functors that take values in chain complexes concentrated in degree zero.

The group homomorphism $G_1\times G_2\to G_1$ induces a functor $\mathcal O_{G_1}\to \Gamma_{G_1\times G_2}$.

This functor induces an adjunction of functor categories $$[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}] \leftrightarrows [\mathcal O_{G_1}^{\operatorname{op}}, \mbox{Top}]$$ where the right adjoint is the pullback. The left adjoint, a.k.a the left Kan extension, can be identified with the functor $X\mapsto \frac{X}{1\times G_2}$. Here I have identified the functor categories with spaces with action of $G_1\times G_2$ and of $G_1$ respectively.

Similarly there is a pair of adjoint functors between categories of coefficient systems

$$[\Gamma_{G_1\times G_2}, \mbox{Ch}] \leftrightarrows [\mathcal O_{G_1}, \mbox{Ch}].$$ Here, again, the right adjoint is the pullback, and it is equivalent to the functor $F$ that you describe.

Finally, the Bredon homology groups $H_*^{G_1\times G_2}(X, FN)$ can be identified with the homology groups of the coend $$C_*\left(X^H\right)\otimes_{H\in \Gamma_{G_1\times G_2}} FN$$ and similarly the Bredon homology groups $H_*^{G_1}\left(\frac{X}{1\times G_2}, N\right)$ can be identified with the homology groups of the following coend $$C_*\left(\frac{X}{1\times G_2}^H\right)\otimes_{H\in \mathcal O_{G_1}} N.$$ The equivalence of the two coends follows from the adjunction.

TL; DR: Suppose you have a functor between small categories $\mathcal C_0\to \mathcal C$. Let $\mathcal D$ be a locally small category such as Top or Ch. Then there is an adjunction of functor categories $$ L:[\mathcal C_0, \mathcal D]\leftrightarrows [\mathcal C, \mathcal D]:R$$ Where the right adjoint $R$ is the restriction and $L$ is the left Kan extension. There is a similar adjunction between categories of contravariant functors. Now suppose we have functors $F\colon \mathcal C_0\to \mathcal D$ and $G\colon \mathcal C^{\operatorname{op}}\to \mathcal D$. Then there is an isomorphism of coends: $$ F\otimes_{\mathcal C_0} RG\cong LF \otimes_{\mathcal C} G.$$

The isomorphism you ask about is an example of this adjunction.

------ Now with more details -------

By Elmendorf-type theorem, you can identify the category of $G_1\times G_2$-spaces that are $G_2$-free with the category of functors $[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}]$, where Top is the category of pointed spaces. Similarly, you can identify coefficient systems of the kind you consider with functors $[\Gamma_{G_1\times G_2}, \mbox{Ch}]$, where Ch denotes the category of chain complexes. More precisely, coefficient systems correspond to functors that take values in chain complexes concentrated in degree zero.

The group homomorphism $G_1\times G_2\to G_1$ induces a functor $\mathcal O_{G_1}\to \Gamma_{G_1\times G_2}$.

This functor induces an adjunction of functor categories $$[\Gamma_{G_1\times G_2}^{\operatorname{op}}, \mbox{Top}] \leftrightarrows [\mathcal O_{G_1}^{\operatorname{op}}, \mbox{Top}]$$ where the right adjoint is the pullback. The left adjoint, a.k.a the left Kan extension, can be identified with the functor $X\mapsto \frac{X}{1\times G_2}$. Here I have identified the functor categories with spaces with action of $G_1\times G_2$ and of $G_1$ respectively.

Similarly there is a pair of adjoint functors between categories of coefficient systems

$$[\Gamma_{G_1\times G_2}, \mbox{Ch}] \leftrightarrows [\mathcal O_{G_1}, \mbox{Ch}].$$ Here, again, the right adjoint is the pullback, and it is equivalent to the functor $F$ that you describe.

Finally, the Bredon homology groups $H_*^{G_1\times G_2}(X, FN)$ can be identified with the homology groups of the coend $$C_*\left(X^H\right)\otimes_{H\in \Gamma_{G_1\times G_2}} FN$$ and similarly the Bredon homology groups $H_*^{G_1}\left(\frac{X}{1\times G_2}, N\right)$ can be identified with the homology groups of the following coend $$C_*\left(\frac{X}{1\times G_2}^H\right)\otimes_{H\in \mathcal O_{G_1}} N.$$ The equivalence of the two coends follows from the adjunction.