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The approach here to the kernel of $u_{X^{\text{an}}}$ is a little different, using Hironaka's Resolution of Singularities.

Proof. By Hironaka's Resolution of Singularities Theorem, there exists a projective, birational morphism, $$(\nu,\nu^\#):(\widetilde{X},\mathcal{O}_{\widetilde{X}})\to (X,\mathcal{O}_X),$$ and a smooth closed subscheme, $$(\widetilde{j},\widetilde{j}^\#):(\widetilde{Y},\mathcal{O}_{\widetilde{Y}})\to (\widetilde{X},\mathcal{O}_{\widetilde{X}}),$$ with ideal sheaf $\widetilde{\mathcal{I}}$ such that $\nu_*\widetilde{\mathcal{I}}\cdot \mathcal{O}_X$ equals $\mathcal{I}$.

Since $X$ is already smooth, the natural homomorphism $\nu^\#:\mathcal{O}_X[0]\to R\nu_*\mathcal{O}_{\widetilde{X}}$ is a quasi-isomorphism compatible with arbitrary base change, including base change to Artinian, local $\mathbb{C}$-schemes in $X$. This is enough to conclude that also $\mathcal{O}_{X^{\text{an}}}[0]\to R\nu^{\text{an}}_*\mathcal{O}_{\widetilde{X}^{\text{an}}}$ is a quasi-isomorphism. Thus, the homomorphism of sheaves of $\mathbb{C}$-algebras, $\mathcal{O}_{X^\text{an}} \to \nu_*\mathcal{O}_{\widetilde{X}}^\text{an}$, is an isomorphism. Together with the previous paragraph, it follows that the natural homomorphism from $\mathcal{O}_{X^\text{an}}/\mathcal{I}\cdot \mathcal{O}_{X^\text{an}}$ to the pushforward of $C^0_{\widetilde{Y}^\text{an}}(\mathbb{C})$ is injective. This homomorphism factors through the natural homomorphism, $$\mathcal{O}_{X^\text{an}}/\mathcal{I}\cdot \mathcal{O}_{X^\text{an}} \to j^{\text{an}}_*C^0_{Y^\text{an}}(\mathbb{C}).$$ Thus, also the natural homomorphism is injective. QED

The approach here to the kernel of $u_{X^{\text{an}}}$ is different, using Hironaka's Resolution of Singularities.

Proof. By Hironaka's Resolution of Singularities Theorem, there exists a projective, birational morphism, $$(\nu,\nu^\#):(\widetilde{X},\mathcal{O}_{\widetilde{X}})\to (X,\mathcal{O}_X),$$ and a smooth closed subscheme, $$(\widetilde{j},\widetilde{j}^\#):(\widetilde{Y},\mathcal{O}_{\widetilde{Y}})\to (\widetilde{X},\mathcal{O}_{\widetilde{X}}),$$ with ideal sheaf $\widetilde{\mathcal{I}}$ such that $\nu_*\widetilde{\mathcal{I}}\cdot \mathcal{O}_X$ equals $\mathcal{I}$. By Serre's GAGA, also $\nu^{\text{an}}_*\widetilde{\mathcal{I}}^{\text{an}}$ equals $(\nu_*\widetilde{\mathcal{I}})^{\text{an}}$, so that the same result holds for the associated analytic spaces.

Since $X$ is already smooth, the natural homomorphism $\nu^\#:\mathcal{O}_X[0]\to R\nu_*\mathcal{O}_{\widetilde{X}}$ is a quasi-isomorphism compatible with arbitrary base change. In particular, $\mathcal{O}_X\to \nu_*\mathcal{O}_{\widetilde{X}}$ is an isomorphism. By GAGA, also $\mathcal{O}_X^{\text{an}} \to \nu_*^{\text{an}}\mathcal{O}^{\text{an}}_{\widetilde{X}}$ is an isomorphism. Together with the previous paragraph, it follows that the natural homomorphism from $\mathcal{O}_{X^\text{an}}/\mathcal{I}\cdot \mathcal{O}_{X^\text{an}}$ to the pushforward of $C^0_{\widetilde{Y}^\text{an}}(\mathbb{C})$ is injective. This homomorphism factors through the natural homomorphism, $$\mathcal{O}_{X^\text{an}}/\mathcal{I}\cdot \mathcal{O}_{X^\text{an}} \to j^{\text{an}}_*C^0_{Y^\text{an}}(\mathbb{C}).$$ Thus, also this second natural homomorphism is injective. QED

Edit. I added some lemmas to address the issue raised by David Speyer. The books by Grauert and Grauert-Remmert are wonderful sources. The proofs in those books are the "correct" arguments, using "sledgehammers" as little as possible. Even though it is a sledgehammer, Hironaka's Resolution of Singularities Theorem does quickly establish the result.

For every locally finite type $\mathbb{C}$-scheme $(X,\mathcal{O}_X)$, denote by $(i_X,i_X^\#):(X^\text{an},\mathcal{O}_X^{\text{an}})\to (X,\mathcal{O}_X)$ the associated complex analytic space. For every topological space $S$, denote by $C^0_S(\mathbb{C})$ the sheaf of continuous, $\mathbb{C}$-valued functions on $S$. For every complex analytic space $(S,\mathcal{O}_S)$, denote by $u_S:\mathcal{O}_S\to C^0_S(\mathbb{C})$ the natural homomorphism of sheaves of $\mathbb{C}$-algebras.

Edit. I added some lemmas to address the issue raised by David Speyer and the OP. The books by Grauert and Grauert-Remmert are wonderful sources. The proofs in those books are the "correct" arguments, using "sledgehammers" as little as possible. Even though it is a sledgehammer, Hironaka's Resolution of Singularities Theorem does quickly establish the result. Also, I will also use the smaller sledgehammer ("club hammer"?) of GAGA.

For every locally finite type $\mathbb{C}$-scheme $(X,\mathcal{O}_X)$, denote by $(i_X,i_X^\#):(X^\text{an},\mathcal{O}_X^{\text{an}})\to (X,\mathcal{O}_X)$ the associated complex analytic space. For every topological space $S$, denote by $C^0_S(\mathbb{C})$ the sheaf of continuous, $\mathbb{C}$-valued functions on $S$. For every complex analytic space $(S,\mathcal{O}_S)$, denote by $u_S:\mathcal{O}_S\to C^0_S(\mathbb{C})$ the natural homomorphism of sheaves of $\mathbb{C}$-algebras.

Question 0. For a finite type $\mathbb{C}$-scheme $(X,\mathcal{O}_X)$, what is the kernel of $u_{X^{\text{an}}}:\mathcal{O}_X^{\text{an}}\to C^0_{X^{\text{an}}}(\mathbb{C})$?

For the nilradical $\mathcal{N}\subset \mathcal{O}_X$, certainly $\mathcal{N}\cdot \mathcal{O}_X^{\text{an}}$ is contained in the kernel. Most of the comments for the question and for this answer focus on the problem of proving that the kernel equals $\mathcal{N}\cdot \mathcal{O}_X^{\text{an}},$ which it does. One approach uses "analytic reducedness" of the local rings $\mathcal{O}_{X,p}/\mathcal{N}_p$ at $\mathbb{C}$-points $p\in X^{\text{an}}$. Not all reduced local Noetherian rings are analytically reduced. However, excellent local rings that are reduced are analytically reduced, cf. the Wikipedia page for excellent local rings.

https://en.wikipedia.org/wiki/Excellent_ring#Properties

The approach here to the kernel of $u_{X^{\text{an}}}$ is a little different, using Hironaka's Resolution of Singularities.

For every locally finite type $\mathbb{C}$-scheme $(X,\mathcal{O}_X)$, denote by $(i_X,i_X^\#):(X^\text{an},\mathcal{O}_X^{\text{an}})\to (X,\mathcal{O}_X)$ the associated complex analytic space. For every topological space $S$, denote by $C^0_S(\mathbb{C})$ the sheaf of continuous, $\mathbb{C}$-valued functions on $S$.

Lemma 1. For every finite type $\mathbb{C}$-scheme, the associated complex analytic space is a complex manifold if and only if $(X,\mathcal{O}_X)$ is a smooth $\mathbb{C}$-scheme. In particular, if $(X,\mathcal{O}_X)$ is a smooth $\mathbb{C}$-scheme, then we have injectivity of the natural homomorphism of sheaves of $\mathbb{C}$-algebras from $\mathcal{O}_X^{\text{an}}$ to $C^0_{X^\text{an}}(\mathbb{C})$.

Lemma 4. For every finite type, affine $\mathbb{C}$-scheme $(X,\mathcal{O}_X)$, for every surjection of coherent $\mathcal{O}_X$-sheaves, $\phi:\mathcal{F}\to \mathcal{G}$, the induced map $\phi^{\text{an}}(X^{\text{an}}):\mathcal{F}^{\text{an}}(X^{\text{an}})\to \mathcal{G}^{\text{an}}(X^{\text{an}})$ is surjective.

Fix a finite generating set $f_1,\dots,f_r$ of the ideal $I = \mathcal{I}(X)$. By Lemma 4, the induced map, $$(\mathcal{O}_X^{\text{an}})(X^\text{an})^{\oplus r} \to (\mathcal{I}\cdot\mathcal{O}_X^{\text{an}})(X^\text{an}), \ \ (u_1,\dots,u_r)\mapsto u_1f_1 + \dots + u_rf_r,$$ is surjective. By construction, the image is in $I^{\text{hol}}$. Therefore $I^{\text{hol}}$ equals all of $(\mathcal{I}\cdot\mathcal{O}_X^{\text{an}})(X^\text{an})$. Thus, $g$ is an element of $I^{\text{hol}}$. QED

For every locally finite type $\mathbb{C}$-scheme $(X,\mathcal{O}_X)$, denote by $(i_X,i_X^\#):(X^\text{an},\mathcal{O}_X^{\text{an}})\to (X,\mathcal{O}_X)$ the associated complex analytic space. For every topological space $S$, denote by $C^0_S(\mathbb{C})$ the sheaf of continuous, $\mathbb{C}$-valued functions on $S$. For every complex analytic space $(S,\mathcal{O}_S)$, denote by $u_S:\mathcal{O}_S\to C^0_S(\mathbb{C})$ the natural homomorphism of sheaves of $\mathbb{C}$-algebras.

Lemma 1. For every finite type $\mathbb{C}$-scheme, the associated complex analytic space is a complex manifold if and only if $(X,\mathcal{O}_X)$ is a smooth $\mathbb{C}$-scheme. In particular, if $(X,\mathcal{O}_X)$ is a smooth $\mathbb{C}$-scheme, then the homomorphism $u_{X^\text{an}}$ is injective.

Lemma 4. For every finite type $\mathbb{C}$-scheme $(Y,\mathcal{O}_Y)$, the scheme is reduced if and only if the homomorphism $u_{Y^{\text{an}}}$ is injective. In particular, for the nilradical $\mathcal{N}\subset \mathcal{O}_Y$, the nilradical of $\mathcal{O}_{Y}^\text{an}$ equals $\mathcal{N}\cdot \mathcal{O}_Y^\text{an}$.

Proof. Since $\mathcal{O}_Y^\text{an}$ is flat over $i_Y^{-1}\mathcal{O}_Y$, the nilradical of $\mathcal{O}_Y^\text{an}$ contains $\mathcal{N}\cdot \mathcal{O}_Y^\text{an}$. Thus, if $(Y,\mathcal{O}_Y)$ is nonreduced, then $u_{Y^\text{an}}$ is not injective.

Conversely, assume that $(Y,\mathcal{O}_Y)$ is reduced. To prove that $u_{Y^{\text{an}}}$ is injective and that $\mathcal{N}\cdot \mathcal{O}_Y^{\text{an}}$ equals the entire nilradical, it suffices to work locally. Locally there are closed immersions of $(Y,\mathcal{O}_Y)$ into affine space. Thus, the result follows from the previous lemma. QED

Lemma 5. For every finite type, affine $\mathbb{C}$-scheme $(X,\mathcal{O}_X)$, for every surjection of coherent $\mathcal{O}_X$-sheaves, $\phi:\mathcal{F}\to \mathcal{G}$, the induced map $\phi^{\text{an}}(X^{\text{an}}):\mathcal{F}^{\text{an}}(X^{\text{an}})\to \mathcal{G}^{\text{an}}(X^{\text{an}})$ is surjective.

Fix a finite generating set $f_1,\dots,f_r$ of the ideal $I = \mathcal{I}(X)$. By Lemma 5, the induced map, $$(\mathcal{O}_X^{\text{an}})(X^\text{an})^{\oplus r} \to (\mathcal{I}\cdot\mathcal{O}_X^{\text{an}})(X^\text{an}), \ \ (u_1,\dots,u_r)\mapsto u_1f_1 + \dots + u_rf_r,$$ is surjective. By construction, the image is in $I^{\text{hol}}$. Therefore $I^{\text{hol}}$ equals all of $(\mathcal{I}\cdot\mathcal{O}_X^{\text{an}})(X^\text{an})$. Thus, $g$ is an element of $I^{\text{hol}}$. QED