The question is essentially already answered in the comments and answers, but I thought it might be valuable to combine these observations into something definitive.

Solovay's model is not unique. The model produced by Solovay's construction depends on the choice of ground model, and on the choice of generic filter. We can always arrange that the ground model satisfies the axiom of constructibility, and when this is the case, it is possible to establish many familiar theorems in the resulting Solovay model by appealing to absoluteness. This is discussed in my paper V*-algebras. There, I work in the Chang model, but as I explain in the first paragraph of section 5, all the results that I state in the Chang model are also true in a Solovay model of the kind I just described, except for the axiom of determinacy.

In such a Solovay model, all states on $\mathcal B(\mathcal H)$ are normal, whenever $\mathcal H$ is separable (remark 8.41). If $\mathcal H = \ell^2([0,1])$, then the state $\lambda\colon x\mapsto \int_0^1\langle e_t| x e_t\rangle dt$ is not normal, as jjcale suggested, because the projections onto the subspaces $\mathbb C e_t$ sum to the identity, but $\lambda$ vanishes on each of these projections. Assuming sufficient large cardinal axioms, the Chang model satisfies the axiom of determinacy, so $\omega_1$ is a measurable cardinal, which yields a non-normal state on $\mathcal B(\mathcal \ell^2(\omega_1))$, as Ashutosh suggested. This state is even less normal than $\lambda$, in the sense that $\lambda$ is normal for well-ordered sequences, but this state is obviously not.

This answers the question just for the Solovay models obtained from ground models satisfying the axiom of constructibility. I would guess that this answer is also correct for the other Solovay models.

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