I believe that in your situation, $B$ indeed has an isolated singularity at the maximal ideal $\mathfrak{n} \subseteq B$. Let me first give two possible definitions for “isolated singularity”; please let me know if there are standard definitions for these notions, and I will edit this answer accordingly!

**Definition.** Let $X$ be a locally noetherian scheme.

We say that $X$ has an *isolated singularity* at a closed point $x \in X$ if there is an open neighborhood $U \ni x$ such that the scheme $U \smallsetminus \{x\}$ is regular.

Suppose that $X$ is a scheme over a field $k$. We say that $X$ has a *geometric isolated singularity* at a closed point $x \in X$ if there is an open neighborhood $U \ni x$ such that the scheme $U \smallsetminus \{x\}$ is geometrically regular over $k$.

Adopting either definition, the ring $B$ in your situation has a isolated singularity (resp. geometric isolated singularity) at the maximal ideal $\mathfrak{n} \subseteq B$. We can in fact prove the following more general result:

**Proposition.** *Let $X$ be a locally noetherian scheme (resp. locally noetherian scheme over a field $k$) with an isolated singularity (resp. geometric isolated singularity) at a closed point $x \in X$, such that $\mathcal{O}_{X,x}$ is a $G$-ring. Then, $\operatorname{Spec}\widehat{\mathcal{O}}_{X,x}$ has an isolated singularity (resp. geometric isolated singularity) at the unique closed point $\widehat{x}$.*

Note that this proposition applies to your situation, since $A$ is of fintie type over a field, hence a $G$-ring in the sense of [Mat89, p. 256] by [Mat89, Cor. to Thm. 32.6] ($A$ is also excellent, but we only need the $G$-ring part of the definition of excellence).

*Proof.* Let $U \subseteq \operatorname{Spec} \mathcal{O}_{X,x}$ be the intersection of $\operatorname{Spec} \mathcal{O}_{X,x}$ and a neighborhood of $x$ in $X$ satisfying the condition in the definition above. Let $f\colon \operatorname{Spec} \widehat{\mathcal{O}}_{X,x} \to \operatorname{Spec} \mathcal{O}_{X,x}$ be the morphism induced by the completion homomorphism. The morphism
$$f^{-1}(U) \smallsetminus \{\widehat{x}\} \longrightarrow U \smallsetminus \{x\}\tag{1}\label{eq:completion}$$
is regular in the sense of [Mat89, p. 255] since the morphism $f$ is regular by the $G$-ring condition, and regular morphisms are preserved under base change [EGAIV$_2$, Prop. 6.8.3(iii)].

For isolated singularities, we apply [Mat89, Thm. 23.7] to the (localizations of the) morphism \eqref{eq:completion} to conclude that $f^{-1}(U) \smallsetminus \{\widehat{x}\}$ is regular.

For geometric isolated singularities, the composition
$$f^{-1}(U) \smallsetminus \{\widehat{x}\} \longrightarrow U \smallsetminus \{x\} \longrightarrow \operatorname{Spec} k$$
is regular by [Mat89, Thm. 32.1], i.e., $f^{-1}(U) \smallsetminus \{\widehat{x}\}$ is geometrically regular over $k$. $\blacksquare$

### References

[EGAIV$_2$] Alexander Grothendieck and Jean Dieudonné. “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II.” *Inst. Hautes Études Sci. Publ. Math.* (1965), no. 24, 1–231. DOI: 10.1007/BF02684322. MR: 199181.

[Mat89] Hideyuki Matsumura. *Commutative ring theory.* Second ed. Translated from the Japanese by M. Reid. Cambridge Stud. Adv. Math. 8. Cambridge Univ. Press, Cambridge, 1989. DOI: 10.1017/CBO9781139171762. MR: 1011461.