Have best options on plan to get on the ground?
Aviate, Navigate, Communicate
Autopilot should handle the AVIATing, but the captain (or 1st officer if that becomes a delegated duty) must give this sufficient, multi-tasked attention to assure it's not been compromised, presumably related to the smoke emergency, and keeps the a/c upright and (NAVIGATE) flying toward the selected nearest acceptable airport on a reasonable altitude profile.
If I had just diverted 239 people, a $250M airplane, and my own little behind to a somewhat remote SE Asian island, I think it prudent that the captain (while he's immediately giving full attention to finding the source of the problem) let the non-flying pilot (1st officer, usually) give six seconds of attention to COMMUNICATing with ATC to have them get us sufficient rescue materiel on site to handle us when we "arrived".
"Center, 370 Heavy, May-Day, May-Day, diverting nearest due for flight deck smoke of unknown origin" (and listen for the acknowledgement, only then turning full attention to resolve the emergency or other assigned duties). Let ATC do the work to simulate what my NAV equipment is going to tell me is the nearest acceptable runway and follow my descent. I'd talk to them again only if I deem I have time.
HF