Airline captain here.

We definitely involve the dispatcher in the diversion decision. Especially if it's an unplanned diversion, where the big-picture view the dispatcher has is very useful for us in our metal tube.

Sure, company dispatchers are usually part of the conversation, and in non-emergency diversions (i.e. the vast majority), they may suggest specific airports that would be more convenient for company logistics. But the final decision is always the pilots' - and once they've declared an emergency, more or less every single airfield, including military, becomes available to them.

Unless they are in an emergency and are busy with aviating, they will coordinate with their dispatcher on diverting, even if only to verify that the weather at the intended alternate is still favorable. Per the FAA regulations, the PIC and the dispatcher have joint operational control over the flight. Of course, at the end of the day, only the pilots have their hands on the controls, so they can make the plane do what they want—but from a legal standpoint, the dispatcher and pilot-in-command have equal & shared responsibility for the safe operation of the flight.