I agree that this result would benefit from being reformulated using algebraic spaces. Eg show representability by algebraic spaces using Artin's criterion, and then use the fact that a quasi-finite separated morphism of aglebraic spaces is representable by schemes. I feel the same about Raynaud's, Un critère d’effectivité de descente, Th 4.1

Could we not also argue directly that $f$ is flat by the fiberwise criterion for flatness (since $G$ and $G'$ are flat locally of finite presentation over the base)?

In your proof you use that $f_\ast O_X$ is flat. This may not be the case if you only assume that $f$ is flat. This is the case if the geometric fibers are reduced though.

I think this is only true for $R$ and $B$ local rings. To reduce to the local case, we need that $f$ is regular in $B/(n \cap R)B$ for all maximal ideal $n$ of $B$, cf Matsumura, Commutative Ring Theory, Theorem 22.6.

Sorry, I missed your comment! It comes from the fact that all isogenies of degree $m$ starting from $E_1$ will give you an elliptic curve $E_3$ with endomorphism ring of conductor a divisor of $m$, so in particular $\mathrm{End}(E_3) \supset \mathcal{O}$. This means that $E_3$ is rational over $F$, so the isogeny is rational over $F$. In particular the Galois action is of the form $\lambda \mathrm{Id}$ on the $m$-torsion; but since we have a rational point $\lambda=1$.

@Yasuda: you can do Gröbner basis over R when you know how to do linear algebra over R (see for instance the survey "Grobner Bases with Coefficients in Rings" by Franz Paue). So in particular when R is an euclidean domain like $\mathbb{Z}$ (of course in practice it will be a lot slower than over a field, and over fields Gröbner basis computation can be quite long already...) At least over $\mathbb{Z}$ it is implemented by Singular (hence Sage) and Magma.