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As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_0,...,i_n}U_{i_0}\times_X\cdots\times_XU_{i_n}$.

A result of Dugger (found in "Topological Hypercovers") shows that geometric realization of this simplicial space is weakly homotopy equivalent to $X$, and so is the homotopy colimit of it. This induces isomorphisms on $h^*$ between the geometric realization of $\check C(p)$ to $X$, and between the homotopy colimit of $\check C(p)$ to $X$.

If $\check C(p)$ were Reedy cofibrant, Theorem 5.83, on page 163 of "Generalized Cohomology" by Kōno and Tamaki, would say there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$. This would be the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_0,...,i_n\}\subseteq I}h^*(U_{i_0}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.

But $\check C(p)$ is not in general Reedy cofibrant (See "Topological Hypercovers" again; the discussion in the beginning of Secion 3). The idea is that Reedy cofibrancy would imply that the finite intersections are cofibrant, which isn't necessarily the case.

One way to recover the result would be to take a cofibrant replacement of $\check C(p)$ first and then take the cohomology spectral sequence of that resulting simplicial space. This is precisely the homotopy colimit spectral sequence, but unfortunately does not have as nice a description at the second page; it involves taking the homology of the Moore normalization of the simplicial replacement.

This generalizes (Dugger's Primer, Prop. 18.17): given a spectrum $\mathscr E$, and a diagram $F : \mathbf C \to \textbf{Top}$, there is a spectral sequence describd by taking the left derived colimit $$ E_{s, t}^2 = H_s(\mathbf C, \mathscr E_t(F)) = \operatorname{colim}_s \mathscr E_t(F) \Rightarrow \mathscr E_{s + t}(\operatorname{hocolim}_{\mathbf C}(F)) \cong \mathscr E_{s + t}(X). $$

Here, we would choose the diagram to be the above simplicial space, and the desired K-Theory spectrum.

As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_0,...,i_n}U_{i_0}\times_X\cdots\times_XU_{i_n}$.

A result of Dugger (found in "Topological Hypercovers") shows that geometric realization of this simplicial space is weakly homotopy equivalent to $X$, and so is the homotopy colimit of it. This induces isomorphisms on $h^*$ between the geometric realization of $\check C(p)$ to $X$, and between the homotopy colimit of $\check C(p)$ to $X$.

If $\check C(p)$ were Reedy cofibrant, Theorem 5.83, on page 163 of "Generalized Cohomology" by Kōno and Tamaki, would say there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$. This would be the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_0,...,i_n\}\subseteq I}h^*(U_{i_0}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.

But $\check C(p)$ is not in general Reedy cofibrant (See "Topological Hypercovers" again; the discussion in the beginning of Secion 3). The idea is that Reedy cofibrancy would imply that the finite intersections are cofibrant, which isn't necessarily the case.

One way to recover the result would be to take a cofibrant replacement of $\check C(p)$ first and then take the cohomology spectral sequence of that resulting simplicial space. This is precisely the homotopy colimit spectral sequence, but unfortunately does not have as nice a description at the second page; it involves taking the homology of the Moore normalization of the simplicial replacement.

This generalizes (Dugger's Primer, Prop. 18.17): given a spectrum $\mathscr E$, and a diagram $F : \mathbf C \to \textbf{Top}$, there is a spectral sequence described by taking the left derived colimit $$ E_{s, t}^2 = H_s(\mathbf C, \mathscr E_t(F)) = \operatorname{colim}_s \mathscr E_t(F) \Rightarrow \mathscr E_{s + t}(\operatorname{hocolim}_{\mathbf C}(F)) \cong \mathscr E_{s + t}(X). $$

Here, we would choose the diagram to be the above simplicial space, and the desired K-Theory spectrum.

As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_1,...,i_n}U_{i_1}\times_X\cdots\times_XU_{i_n}$.

A result of Dugger (found in "Topological Hypercovers") shows that geometric realization of this simplicial space is weakly homotopy equivalent to $X$, and so is the homotopy colimit of it. This induces isomorphisms on $h^*$ between the geometric realization of $\check C(p)$ to $X$, and between the homotopy colimit of $\check C(p)$ to $X$.

If $\check C(p)$ were Reedy cofibrant, Theorem 5.83, on page 163 of "Generalized Cohomology" by Kōno and Tamaki, would say there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$. This would be the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_1,...,i_n\}\subseteq I}h^*(U_{i_1}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.

But $\check C(p)$ is not in general Reedy cofibrant. One way to recover the result would be to take a cofibrant replacement of $\check C(p)$ first and then take the cohomology spectral sequence of that resulting simplicial space. This is precisely the homotopy colimit spectral sequence, but unfortunately does not have as nice a description at the second page; it involves taking the homology of the Moore normalization of the simplicial replacement.

This generalizes (Dugger's Primer, Prop. 18.17): given a spectrum $\mathscr E$, and a diagram $F : \mathbf C \to \textbf{Top}$, there is a spectral sequence describd by taking the left derived colimit $$ E_{s, t}^2 = H_s(\mathbf C, \mathscr E_t(F)) = \operatorname{colim}_s E_t(F) \Rightarrow E_{s + t}(\operatorname{hocolim}_{\mathbf C}(F)) \cong E_{s + t}(X). $$

Here, we would choose the diagram to be the above simplicial space, and the desired K-Theory spectrum.

As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_0,...,i_n}U_{i_0}\times_X\cdots\times_XU_{i_n}$.

A result of Dugger (found in "Topological Hypercovers") shows that geometric realization of this simplicial space is weakly homotopy equivalent to $X$, and so is the homotopy colimit of it. This induces isomorphisms on $h^*$ between the geometric realization of $\check C(p)$ to $X$, and between the homotopy colimit of $\check C(p)$ to $X$.

If $\check C(p)$ were Reedy cofibrant, Theorem 5.83, on page 163 of "Generalized Cohomology" by Kōno and Tamaki, would say there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$. This would be the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_0,...,i_n\}\subseteq I}h^*(U_{i_0}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.

But $\check C(p)$ is not in general Reedy cofibrant (See "Topological Hypercovers" again; the discussion in the beginning of Secion 3). The idea is that Reedy cofibrancy would imply that the finite intersections are cofibrant, which isn't necessarily the case. 

One way to recover the result would be to take a cofibrant replacement of $\check C(p)$ first and then take the cohomology spectral sequence of that resulting simplicial space. This is precisely the homotopy colimit spectral sequence, but unfortunately does not have as nice a description at the second page; it involves taking the homology of the Moore normalization of the simplicial replacement.

This generalizes (Dugger's Primer, Prop. 18.17): given a spectrum $\mathscr E$, and a diagram $F : \mathbf C \to \textbf{Top}$, there is a spectral sequence describd by taking the left derived colimit $$ E_{s, t}^2 = H_s(\mathbf C, \mathscr E_t(F)) = \operatorname{colim}_s \mathscr E_t(F) \Rightarrow \mathscr E_{s + t}(\operatorname{hocolim}_{\mathbf C}(F)) \cong \mathscr E_{s + t}(X). $$

Here, we would choose the diagram to be the above simplicial space, and the desired K-Theory spectrum.

As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_1,...,i_n}U_{i_1}\times_X\cdots\times_XU_{i_n}$.

Under certain (I believe mild) conditions the induced map from its geometric realization to $X$ induces isomorphism on $h^*$. On the other hand - see e. g. Theorem 5.83 on page 163 of "Generalized Cohomology" by Kōno and Tamaki - there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$.

In our case - yet again under some conditions - the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_1,...,i_n\}\subseteq I}h^*(U_{i_1}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.

As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_1,...,i_n}U_{i_1}\times_X\cdots\times_XU_{i_n}$.

A result of Dugger (found in "Topological Hypercovers") shows that geometric realization of this simplicial space is weakly homotopy equivalent to   $X$, and so is the homotopy colimit of it. This induces isomorphisms on $h^*$ between the geometric realization of $\check C(p)$ to $X$, and between the homotopy colimit of $\check C(p)$ to $X$.

If $\check C(p)$ were Reedy cofibrant, Theorem 5.83, on page 163 of "Generalized Cohomology" by Kōno and Tamaki, would say there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$. This would be the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_1,...,i_n\}\subseteq I}h^*(U_{i_1}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.

But $\check C(p)$ is not in general Reedy cofibrant. One way to recover the result would be to take a cofibrant replacement of $\check C(p)$ first and then take the cohomology spectral sequence of that resulting simplicial space. This is precisely the homotopy colimit spectral sequence, but unfortunately does not have as nice a description at the second page; it involves taking the homology of the Moore normalization of the simplicial replacement.

This generalizes (Dugger's Primer, Prop. 18.17): given a spectrum $\mathscr E$, and a diagram $F : \mathbf C \to \textbf{Top}$, there is a spectral sequence describd by taking the left derived colimit $$ E_{s, t}^2 = H_s(\mathbf C, \mathscr E_t(F)) = \operatorname{colim}_s E_t(F) \Rightarrow E_{s + t}(\operatorname{hocolim}_{\mathbf C}(F)) \cong E_{s + t}(X). $$

Here, we would choose the diagram to be the above simplicial space, and the desired K-Theory spectrum.