The number of states of local automata
A late answer, but the bound on synchronization delay has been studied for several classes of automata: see for instance Unambiguous Automata; Béal et al. MCS'08.

In particular; there is a family of deterministic automata that have delay $\Omega(|Q|^2)$ as showed in On the Bound of the Synchronization Delay of a Local Automaton; Béal et al. TCS'98, which matches a corresponding $O(|Q|^2)$ upper bound.

P.S. the synchronization delay defined in the paper is the minimal $k$ for which the deterministic local automaton is $k$-local.