B757 Captain reported loss of both packs caused the loss of cabin pressure. The flight crew requested priority handling and descended to 10;000 feet. The flight crew elected to divert and make a precautionary landing.

Narrative

Aircraft X; Date.At cruise; FL370 flight deck was unusually cold despite repeated attempts to warm it up. Per flight deck cold checklist; tried to stabilize all temp zones by turning trim air off. First try it worked for about 30 minutes. Then got cold again. Tried trim air off again and no change in flight deck. Continued with 2nd checklist; cabin air cold. Turned left pack to STBY N. Then turned right pack to STBY N. Right pack shut down; followed by right ENG bleed off; turned right pack back to auto; no change. Turned left pack back to auto and left pack shut down.We immediately felt the effects of the loss of pressurization and began an emergency descent to 15;000 ft. I flew the airplane initially to begin the descent; directed the First Officer (FO) to don his mask and [request priority handling].FO ran the rapid depressurization/cabin altitude QRC checklist followed by non normals. We transferred control of the aircraft at some point during the descent before leveling off at 10;000 ft. I resumed pilot monitoring (PM) duties while FO flew and spoke with ATC. I ran the following checklists: L and R PACK OFF. We decided to turn back to ZZZ as this was the nearest suitable airport.I contacted the flight attendants to confirm they were OK; then called dispatch and Maintenance Control.Maintenance Control verified the status of the aircraft and that all appropriate checklist were accomplished. I spoke with Flight Attendants (FAs) again to provide information and confirmed everyone was OK.Due to the fact I was conducting IOE with a new hire; I felt that I may have taken longer than normal to initiate contact with the flight attendants and passengers; which was discussed during de-brief. Flight Attendants stated they understood and managed what they needed to do to prepare the cabin for the divert.Purser advised there was a Company Pilot in the cabin; who had offered to come up to assist. Initially I declined; but after completing all checklists and briefing for the approach and landing; I felt it would be a good idea to have a third set of eyes as we were both exhausted and possibly impacted by the [priority] descent. I used Captain's emergency authority and allowed the pilot to access the flight deck.ERRORS.Items missed after detailed de-briefing.Diversion QRG (Quick Reference Guide).Requesting runway data for landing at divert airport.The loss of the right pack when moved to STBY N caused a reaction to return the pack back to the position it was in prior to shutting down. The same action I took for the left pack; going back to AUTO; may have contributed to the total loss of pressurization. Had I left it in the STBY N position we may not have lost the left pack and been able to continue on.I reviewed the checklist post flight and determined that it was an error to have moved the pack switch back to AUTO. Doing an ENG BLEED reset on the right ENG BLEED may also have corrected the initial problem. I was not aware of the error until we had lost pressurization and needed to descend. Regardless of my error; there was definitely a problem with the auto-controller; valve disagreement; and other unknown issues. I did speak with Maintenance Control today and they had yet to determine the underlying problem. I recognize that flying an all-nighter is a high threat and adding IOE to the flight increases the threat. My FO did an excellent job despite being straight out of new-hire training. I also recognize that my troubleshooting abilities were impacted by the late hour.