There is no geometric morphism $f : E \to F$ if $E$ is a realizaiblity topos and $F$ is a Grothendieck topos. Indeed, if we had such an $f$, then for any indexing set $I$, the topos $E$ would have the $I$-fold coproduct $\coprod_I 1$ of terminal objects: such a coproduct exists in $F$ and $f^*$ will carry it over to $E$, as it preserves the terminal object and coproducts. Taking $I$ to be the underlying pca of $E$ shows that this is impossible (by a cadinality argument, see also the notion of width in Lemma 2.11 in [Sheaf toposes for realizability](https://doi.org/10.1007/s00153-008-0090-6), [PDF](https://math.andrej.com/asset/data/sheaves_realizability.pdf) here. Come to think of it, the paper might be of some independent interst to you.

Regarding the opposite direction, I once heard Peter Johstone state that every geometric morphism from a Grothendieck topos to a realizability topos factors through $(\Gamma \dashv \nabla) : \mathsf{Set} \to \mathsf{RT}(A)$, but I cannot remember why (I was too young to understand much of Peter Johstone anyway).

A topos that combines computability and topology in a particularly nice way is the Kleene-Vesley topos $\mathsf{RT}(\mathbb{N}^\mathbb{N}, (\mathbb{N}^\mathbb{N})_\mathsf{eff}$, and a slightly less nice one is $\mathsf{RT}(\mathcal{P}(\mathbb{N}, \mathsf{RE})$, see my [PhD thesis](https://math.andrej.com/2000/09/20/the-realizability-approach-to-computable-analysis-and-topology/) or these [unfinished notes on realizability](https://github.com/andrejbauer/notes-on-realizability).