Homotopically equivalent to Čech nerve?

There's a proof of this fact (in the category of CW-complexes, at least) in section 4.G of Hatcher's "Algebraic Topology." The basic idea is to construct an explicit map $X\to N(\mathfrak{U})$, then use Whitehead's theorem to show that it's a homotopy equivalence. The proof uses paracompactness, but it's only used to construct a partition of unity; the local finiteness condition gives you that for free.

Here's a sketch using the language of homotopy colimits. Hatcher uses a slight variation of this using $\Delta$-sets as opposed to simplicial sets, where his geometric realization coincides with the "fat realization."

Let $\mathscr U = \{U_j\}_{j \in J}$ be an open cover $X$. Suppose that $X$ is paracompact and that all the nonempty finite intersections in $\mathscr U$ are contractible.

Associated to every open cover there is an abstract simplicial complex called the nerve of the cover, written $K = N(\mathscr U)$, that has vertices to be the set $J$, and simplicies the finite subsets $\emptyset \neq I \subseteq J$ such that $U_I \neq \emptyset$. Every simplicial complex can be viewed as a category with objects the set of simplicies and morphisms the inclusions of simplicies.

Define the functors $F, G : K^{\text{op}} \to \textbf{Top}$ as follows. Define $F$ on objects by $F(\sigma) = U_\sigma$ and on morphisms by $\sigma \subseteq \sigma'$ maps to the inclusion $U_{\sigma'} \to U_\sigma$. Define $G$ to be the constant functor on the space with one object $\{*\}$.

Now we can define a natural transformation $\tau : F \Rightarrow G$ by $\tau_\sigma : U_\sigma \to \{*\}$ which by hypothes is a pointwise homotopy equivalence. The natural transformation $\tau$ induces a map on the homotopy colimits, $\tau_* : \operatorname{hocolim} F \to \operatorname{hocolim} G$, which is a homotopy equivalence (since pointwise homotopy equivalence implies homotopy equivalence on the homotopy colimits; for a detailed proof see Theorem 8.3.7, p. 409, Cubical Homotopy Theory by Munson, Volić).

For the first functor, $F$, we have, $$ \operatorname{hocolim} F = \vert \operatorname{srep}_\bullet F \vert = \bigsqcup_{n \geq 0} \bigsqcup_{\sigma_0 \subseteq \cdots \subseteq \sigma_n} U_{\sigma_n} \times \Delta^n / \sim. $$ Dugger [Hypercovers in Topology, p. 6] shows that this is homeomorphic to (the realization of) the ordered Čech complex, $\check C_\bullet^{\text{ord}}(\mathscr U)$ using a neat coordinatization of the barycentric subdivision of the $k$-simpicies, in effect, $$ \Bigg (\bigsqcup_{n \geq 0} \bigsqcup_{\sigma_0 \subseteq \cdots \subseteq \sigma_n} U_{\sigma_n} \times \Delta^n \Bigg) / \sim \quad \cong \quad \Bigg(\bigsqcup_{k \geq 0} \bigsqcup_{j_0 \leq \cdots \leq j_k} U_{j_0, \ldots, j_k} \times \operatorname{sd} \Delta^k \Bigg)/ \sim. $$ The standard argument then, found in Dugger, Hatcher, tom Dieck, shows that $\vert \check C_\bullet^{\text{ord}}(\mathscr U) \vert \simeq X$ when $\mathscr U$ admits partitions of unity, which occurs here since $X$ is paracompact by hypothesis.

For the second functor, $G$, we have $$ \operatorname{hocolim} G = \vert \operatorname{srep}_\bullet G \vert = \vert N_\bullet(K) \vert \cong \vert SS_\bullet(K) \vert \cong \vert N(\mathscr U) \vert. $$ where $N_\bullet(K)$ is the nerve of the category $K$ as a simplicial set, and $SS_\bullet(K)$ is the natural simplicial set associated to every abstract simplicial complex as described in Weibel [Example 8.1.8, p. 258]. The first homeomorphism, $\vert N_\bullet(K) \vert \cong \vert SS_\bullet(K) \vert$, follows from the same subdivision trick used by Dugger. The second follows using the definition of the realization of an abstract simplicial complex, or alternatively, you could take this to be the definition of the realization of a simplicial complex.

Hence, $$ X \simeq \operatorname{hocolim} F \simeq \operatorname{hocolim} G \cong \vert N(\mathscr U) \vert, $$ as desired.