B757 First Officer reported a pack temperature control problem expanded to the loss of both packs and cabin altitude control. The flight crew descended to 10;000 ft. and diverted to the nearest suitable airport.

Narrative

IOE Captain - Pilot Monitoring IOE First officer - Pilot Flying Aircraft X; ZZZ1 - ZZZ2 Divert ZZZ.Cruising at FL370 we had been attempting to control the temperature on the flight deck as it was moving without change to excessively cold. Under the direction of the FM we began to run the (757) Cabin Temperature Cold checklist. We turned off the TRIM AIR switch but this had made the aft cabin excessively hot and so we turned the TRIM AIR switch back to its normal operating position ON. We attempted to use the TRIM AIR switch once more before moving on with the checklist with the same result. We then moved on with the checklist which states PACK control selectors (both) STBY-N; out of an abundance of caution we decided to move one PACK at a time; first the left PACK; this pack had no abnormal effect and was operating properly in STBY-N. Next; after confirmation; we switched the R PACK to STBY-N; immediately this followed with a PACK OFF caution followed again by R Eng Bleed off and an obvious change in airflow. At this point we have L-PACK in STBY-N and R-PACK off (Switch position in STBY-N) before running the PACK OFF (R) non-normal checklist we put both PACKs back into the AUTO position; at this point when L-PACK was put into AUTO; we lost our L-PACK and received a L PACK OFF indication. Immediately after this happened both the Captain and I were physiologically and mentally aware that we needed to don O2 masks and begin a rapid descent; we began both immediately.The Captain assumed control of the aircraft and we then asked for lower and [requested priority handling]; at this point we were both focused on the aircraft state and directed ATC to standby for further communication. We began the [priority] descent and ran the QRC followed by the FM Air system non-normal Cabin Altitude or rapid depressurization or [priority] descent. The cabin was climbing but we were also descending; the FM procedure does direct you to activate the PAX O2 mask but only if greater than 14;000 ft.; when checking this cabin altitude; we were roughly at 11;500 ft. I would say it took roughly 5-7 minutes before cabin alt rose to about 15;000 ft. At no time did I see it climb higher. Initially beginning the descent to 15;000 ft. but shortly followed by a request for 10;000 ft. As a crew we wanted to be certain what the terrain was below; the initial objective was to get down out of 37;000 ft. and make a plan as we got lower. I started the APU during descent as an extra generator and potentially for an additional bleed source as we would come through 17;000 ft.; fuel was considered before making this decision. After completing the checklist; before leveling off; the captain and I switched roles. I was now again the pilot flying; and the Captain began monitoring and coordination. During the descent we had lost our usable frequency for ATC and used 121.5 to get a back in two-way radio communication with ATC. I informed the Captain that at a cabin altitude of 10;000 ft. we were safe to remove our O2 masks but we both decided to keep them on for a little longer in an attempt to mitigate physiological fatigue. At this time the Captain was mainly coordinating with Dispatch; Maintenance; and the Flight Attendants. I was flying the aircraft and taking radios with ATC. We decided on ZZZ as an appropriate alternate based on fuel on board and the additional pilot responsibilities we had to accomplish before landing unpressurized. We got direct ZZZ and decided; with the excess fuel we had; to go down to 8;000 ft. for better oxygen levels for unpressurized flight. With Maintenance on the line; the Captain completed the operational checklist required and was able to do one last coordination with the Flight Attendants. During the Captain's coordination all I was focused on was flying the airplane; managing automation; navigating to ZZZ; and ATC communication. Once we were both back together; we got the ATIS information for ZZZ and began setting up for the approach XXC. During debriefing we found that one thing we did forget to do was pull runway data at ZZZ; I believe this was due to that fact that the weather was CAVOK; runways were greater than 10;000 ft.; and both pilots being familiar with the airport environment and landing requirements. This was missed because of also missing the FOM divert guide.At this time; we were informed of a Company Pilot who was a passenger on our flight. After the immense workload the Captain and I had been under we both decided it would be the safest decision to have a fresh set of eyes to monitor our approach into ZZZ. The Captain used their emergency authority to have the pilot assist us in the flight deck on approach.We had a normal approach and landing in ZZZ and taxied without issue to our gate. Once at the gate we debriefed Maintenance on our situation and outcome. At no time during the flight do I believe the safety of the passengers or crew were at risk. I commend the Captain on their quick thinking; leadership; and experience in this event. I also commend the training team that had recently trained me and prepared me for such an event. The Flight Attendants handled the cabin like calm professionals and were commendable. Much to my surprise there were no reports of any passengers in any sort of physiological or mental distress; we made sure to ask about this in an attempt to coordinate EMS if necessary but everyone checked in as good.