Consider a von Neumann algebra of operators $R$ in a complex generally non-separable Hilbert space $H$ and let $L\subset R$ be the lattice of orthogonal projectors included in $R$. Is $L$ complete?
In other words, if $A\subset L$ is a family with any cardinality, do $\sup A \in L$ and $\inf A \in L$ hold?
Dvurecenskij's book on Gleason's theorem seems to answer positively, but the statement is not clearly written and no references are given. I suspect that in Kadison-Ringrose's volumes there should be the answer, but I cannot check it...
I don't know what proof you know of in the separable case, but every proof I know of works in the inseparable case as well. Here is one such proof.
Let $A\subseteq L$ be any subset. Let $B$ be the set of finite joins of elements of $A$. Let $\sup A$ denote the supremum of $A$ as a set of projections in $B(H)$. It is clear that $\sup B=\sup A$, so it suffices to show that $\sup B\in R$. Since $B$ is a directed set, we can consider the identity map $B\to B(H)$ as a net in $B(H)$. I claim that this net converges to $\sup B$ in the strong operator topology.
To prove this, let us write $P=\sup B$. Note that the image of $P$ is the closure of the union of the images of all the elements of $B$ (we don't have to take finite linear combinations because $B$ is closed under finite joins). In particular, for any $x\in H$ and any $\epsilon>0$, there is some $Q\in B$ and some $y\in H$ such that $\|Px-Qy\|<\epsilon$. But $Qx=QPx$ is the element of $\operatorname{im}(Q)$ that is closest to $Px$, so this implies $\|Px-Qx\|<\epsilon$. For any $R\geq Q$ in $B$, we have $Q\leq R\leq P$, so $\|(P-R)x\|\leq\|(P-Q)x\|<\epsilon$. The existence of a $Q$ such that this is true is exactly what it means for the net to converge in the strong operator topology.
Since $R$ is closed in the strong operator topology, we conclude that $\sup B=\sup A\in R$. It follows that $L$ is complete (and that moreover, the inclusion of $L$ into the lattice of projections in $B(H)$ is a complete homomorphism).