Let $f : E \to E'$ be an isogeny over $\mathbb{Q}$ and let $p$ be a prime of good reduction for $E, E'$.
One way to see that the reductions (of global minimal integral Weierstrass models) $\tilde{E}, \tilde{E'}$ are isogenous over $\mathbb F_p$ is to use Tate modules (which would allow to treat the case of general abelian varieties over number fields, using Tate's isogeny theorem).
The linked answer explains that for $\ell \neq p$, the map $V_{\ell}(f) : V_{\ell}(E) \to V_{\ell}(E')$ is a isomorphism of $G_{\mathbb{Q}}$-representations, with inverse $\frac{1}{\deg(f)} V_{\ell}(f^\vee)$.
If we denote by $r : E \to \tilde{E}$ the reduction mod $p$ map, then statement VII.3.1.b) in Silverman's book "Arithmetic of Elliptic curves" implies that the reduction map induces isomorphisms $E[m] \cong \tilde E[m]$ for all $m$ coprime to $p$, and thus we get an isomorphism $V_{\ell} r : V_{\ell} E \to V_{\ell}(\tilde E)$, and the same holds for $V_{\ell} r' : V_{\ell} E' \to V_{\ell}(\tilde E')$.
We may thus define an map $\overline{V_{\ell} f} : V_{\ell}(\tilde E) \to V_{\ell}(\tilde E')$ so that the following diagram commutes:
[The $T_\ell$ should be $V_\ell$ everywhere.]
Because all other 3 maps are isomorphisms, we see that $\overline{V_{\ell} f}$ is an isomorphism at least of $\mathbb Z_\ell$-modules. To conclude we need to see that $\overline{V_{\ell} f}$ is an isomorphism at least of $G_{\mathbb{F}_p}$-representations.
If we denote by
$D_p \cong
\mathrm{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)
\hookrightarrow
G_{\mathbb{Q}}$
the decomposition subgroup at $p$, we have a surjective morphism $\pi : D_p \twoheadrightarrow G_{\mathbb{F}_p}$.
Then the Galois-equivariance of $\overline{V_{\ell} f}$ follows from the one of $V_{\ell} f$ and $V_{\ell} r, V_{\ell} r'$.
Namely, we have
$$ V_{\ell} r ( \sigma \cdot t ) = \pi(\sigma) \cdot V_{\ell} r(t) $$
whenever $\sigma \in D_p, t \in V_{\ell} E$. This is because on the level of $P =(x,y) \in E[m] \subset E(\overline{\mathbb{Q}})$, we have
$$r(\sigma \cdot (x,y)) = \pi(\sigma) \cdot r(x,y)$$
since $\pi(\sigma) : \bar{x} \mapsto \overline{\sigma(x)}$ is well-defined ($D_p$ is the stabilizer of a prime above $p$ in $\mathcal{O}_{\overline{ \mathbb{Q} }}$).
Note that if $f$ is defined between the minimal integral Weierstrass models of $E,E'$ using a ratio of polynomials having coefficients in $\Bbb Z \setminus p \Bbb Z$, then there is a well-defined reduction $\tilde f : \tilde E \to \tilde E'$ which is a morphism of elliptic curves, and cannot be zero (otherwise $f$ would map all points of $E$ to points having some $p$ in the denominator in the $x,y$-coordinates), hence is an isogeny — so no need of Tate's theorem here in that case.
