One can read on Arveson's 'A short course on Spectral theory', in Remark 2.4.6. Comments on inseparability, regarding a sketch of how to prove the Spectral theorem for inseparable spaces, by following the same way as shown for separable ones:
"If one insists on generalizing this form of the spectral theorem so as to include normal operators acting on inseparable Hilbert spaces, then it is possible to do so but some technical changes are necessary.
The definition of diagonalizable operator must be generalized so as to allow inseparable measure spaces that are not σ-finite. Thus one says that an operator $A \in B(H)$ is diagonalizable if there is a positive measure space $(X, µ)$, a function $f\in L^\infty(X, µ)$, and a unitary operator $W : L^2(X, µ) \to H$ such that $WMf = AW$. One must replace Lemma 2.4.2 with the assertion that the direct sum of a uniformly bounded family $\{A\alpha : \alpha \in I\}$ of diagonalizable operators is diagonalizable, where $I$ is an index set of arbitrary cardinality. The proof of that result is similar to the one given, except that one has to construct uncountable direct sums of measure spaces. This requires some care but poses no substantial difficulties. No change is required for the key Lemma 2.4.4, but one must replace Lemma 2.4.3 with the assertion that every normal operator is a perhaps uncountable direct sum of normal operators having cyclic vectors. Once these preparations are made, the proof of the spectral theorem can be pushed through in general."
As I don't have enough in-depth familiarity with measure theory, I am looking for a good reference where we can follow this proof in all detail perhaps.