"If two units are engaged and a game effect removes one or more trays from one of those units such that those units are no longer engaged, the unit that did not have any trays removed may close in"

Say I've got 2x2 Reanimates frontally engaged with 2x1 spear men and a worm engaged in the rear of the lone spearmen tray. The reanimates attack at initiative 4 and cases the rear spearman tray that's engaged with the worm to be removed.

Does the worm get to close in and attack with its initiative 5 attack?

I do believe that is exactly how it would work.

I think the worm would close in as soon as the tray was removed if it was going to close in.

Yeah, so he would close in as soon as the tray was removed, meaning he'd be able to attack at initiative 5. Even if he were initiative 4, as long as he activated after the Reanimates, he'd still get to close in and then attack.