Here is a straightforward commutative algebra argument. We have to show the following:

**Lemma.** *Let $R$ be a $\mathbf Z$-torsion free ring and $f \colon R[x] \to R[x]$ an étale homomorphism of $R$-algebras. Then $f$ is an isomorphism.*

*Proof.* As noted, this is trivial if $R$ is reduced: then $f'(x) \in R[x]^\times = R^\times$ is constant, so $\deg f = 1$ as $R$ is $\mathbf Z$-torsion free and $\tfrac{\partial}{\partial x}\sum_i a_ix^i = \sum_iia_ix^{i-1}$. We obtain the general case by reduction to the reduced case. Firstly, note that $f$ is defined and étale over some finitely generated subring $A \subseteq R$ (namely the subring generated by the coefficients of $f(x)$ and of $f'(x)^{-1}$), and the result for $f_A \colon A[x] \to A[x]$ implies the result for $f \colon R[x] \to R[x]$. Thus we may assume that $R$ is Noetherian. (Note that a subring of a $\mathbf Z$-torsion-free ring is still $\mathbf Z$-torsion-free.)

Then the radical $I = \operatorname{nil}(R)$ is finitely generated, so satisfies $I^n = 0$ for some $n \in \mathbf Z_{>0}$. Note that $R/I$ is again $\mathbf Z$-torsion free: if $x \in R$ and $m \in \mathbf Z_{>0}$ are such that $mx \in I$, then $m^n x^n = 0$ hence $x^n = 0$ since $R$ is $\mathbf Z$-torsion free. Write $\bar R$ for $R/I$, and $\bar f \colon \bar R[x] \to \bar R[x]$ for the reduction of $f$ modulo $I$.

Then $\bar f$ is still étale, hence an isomorphism since $R$ is reduced. By Nakayama's lemma \[Tag [00DV](https://stacks.math.columbia.edu/tag/00DV)(11)\], we see that $f \colon R[x] \to R[x]$ is surjective since $\bar f$ is. It is also flat and finitely presented, hence isomorphic to the localisation $R[x] \to R[x]_e$ at an idempotent $e \in R[x]$ \[Tag [00U8](https://stacks.math.columbia.edu/tag/00U8)\]. Since $\bar f$ is an isomorphism, we get $e \equiv 1 \pmod I$, i.e. $1-e$ is nilpotent. But $1-e$ is also idempotent, so $1-e=0$ and $e=1$. $\square$

In general, for thickenings $R \twoheadrightarrow R/I$ there is an equivalence of categories between étale $R$-algebras and étale $R/I$-algebras; see \[Tag [0BQB](https://stacks.math.columbia.edu/tag/0BQB)\] for the case of finite étale algebras, and \[Tag [04DZ](https://stacks.math.columbia.edu/tag/04DZ)\] for the general statement (in an even more general setting: thickenings are examples of *universal homeomorphisms*). So for these types of results, you usually only need to think about the reduced case.