Thanks a lot, that nearly clears everything up for me. In your expressions for $\omega_{S}$ and $\omega_{S'}$ presumably you're specializing to certain rational elliptics? Because I believe for general rational elliptic surfaces the fiber product will not be Calabi-Yau.

@FelipeVoloch Thanks for your comment. I'm a little out of my comfort zone here, so maybe I can ask if my idea of twists is correct. A twist $X_{1}^{\sigma}$ should be $X \times \sigma /G$ for a $G$-torsor $\sigma \to \text{Spec}(\mathbb{F}_{p^{n}})$. Is that product and quotient the usual ones, or is there something subtle here? Finally, the sum you write is over $\sigma \in H^{1}(\mathbb{F}_{p^{n}}, G)$, correct?

I see, thanks a lot. If we expand the definition of Jacobi forms to half-integral weight, do you know what the index of the theta functions I write is, as well as which congruence subgroups they transform under? Or perhaps these questions don't quite make sense.

Well because they look schematically like that general form in the first equation of my OP in the case of N=1. And in fact, just after Eichler and Zagier write that formula they mention that it's a generalization of what Jacobi studied for N=1. I figured they were referring to these theta functions.

That second isomorphism in the figure from Eichler and Zagier which I provide seems to be exactly equation (4.23) in DMZ. You expect the conclusion should be that whatever "kind" of Jacobi form you start with, you produce the corresponding kind of modular object under that sequence of isomorphisms?