= Solution
Let $\mathcal I\subseteq\mathcal O_X$ be the <ideal sheaf> of $Y$. The <formal completion of a scheme> along $Y$ has underlying topological space $|Y|$ and topological structure sheaf
$$
\boxed{X_{/Y}=\left(|Y|,\ \varprojlim_{n\geq0}(\mathcal O_X/\mathcal I^{n+1})|_{|Y|}\right).}
$$
The topology is the <adic topology> given by the kernels of the projections to the successive <infinitesimal neighbourhoods> $X_n=(|Y|,\mathcal O_X/\mathcal I^{n+1})$. In the locally <Noetherian scheme> setting of these examples this is a <formal scheme>. On a Noetherian <affine scheme>, with $Y=V(I)\subseteq\operatorname{Spec}A$, it is the <formal spectrum> $\operatorname{Spf}\widehat A$, where $\widehat A=\varprojlim_nA/I^{n+1}$. Thus the completion retains all orders of infinitesimal information normal to $Y$.
Here $Z=D(t)=\operatorname{Spec}k[q,t,t^{-1}]$ is an <open subscheme> of $X$. The relevant affine completions are
$$
X_{/(q=0)}=\operatorname{Spf}\bigl(k[t][[q]]\bigr),\qquad
Z_{/(q=0)}=\operatorname{Spf}\bigl(k[t,t^{-1}][[q]]\bigr),
$$
with the $q$-adic topology. The complement of $t=0$ in the first <formal scheme> is its formal open $D(t)$. The sections on this open are the <completed localization of an adic ring>:
$$
\begin{aligned}
\Gamma(D(t),\mathcal O_{X_{/(q=0)}})
&=\varprojlim_{n\geq0}\bigl(k[t,q]/(q^{n+1})\bigr)[t^{-1}]\\
&=\varprojlim_{n\geq0}k[t,t^{-1},q]/(q^{n+1})
=k[t,t^{-1}][[q]].
\end{aligned}
$$
The same calculation on every smaller <principal open subset> identifies the structure sheaves, not just their global sections. Equivalently, every <infinitesimal neighbourhood> of $Z\cap(q=0)$ is precisely the restriction to $D(t)$ of the corresponding neighbourhood in $X$. By <formal completion commutes with restriction to an open subscheme>,
$$
\boxed{Z_{/(q=0)}\ \cong\ X_{/(q=0)}\setminus(t=0).}
$$
The <isomorphism> is canonical and respects the morphisms to $\operatorname{Spf}k[[q]]$.
It would give the wrong answer to replace the formal-open structure sheaf by the ordinary <localization of a ring> $k[t][[q]][t^{-1}]$. That <ring> has a uniform bound on the negative powers of $t$ across the coefficients of any one element, whereas
$$
\sum_{n\geq0}q^nt^{-n}\ \in\ k[t,t^{-1}][[q]]
$$
has no such bound. This difference between ordinary and completed localization does not obstruct the <isomorphism> of <formal schemes>; it explains why the completed localization is essential.
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