737-800 First Officer reported rising oil temperature on the #1 engine; resulting in an engine shutdown and a single-engine diversion.

Narrative

In cruise returning to the flight deck from a lavatory break; my Captain informed me the oil temperature on the number 1 engine was rising; we were on a vector possibly diverting to ZZZ and ATC was notified. As I got strapped in and began assessing the situation the oil temperature proceeded to turn red and was pinned in the red for over 30 seconds while the Captain already had the high oil temperature QRH procedure out which stated if red proceed to shutdown engine 1 checklist. We noticed the oil quantity dipped as low as 3 qts while descending/diverting and after about 1 minute in the red the oil temperature number indication went blank. We proceeded with the engine shutdown checklist. Notified ATC; Dispatch the Flight Attendants for a precautionary landing and eventually the passengers. We followed all non normal and normal checklist then we landed in ZZZ followed by Airport Firefighting parked all of which was uneventful and smooth. We made the log entry talked with our Chief Pilots; Dispatch and Mechanics while making a log entry of the event. The main factor was the high oil temperature then low oil quantity. Outside of that the rest of the diversion went smoothly. I believe we operated professionally; followed SOPs and made a safe diversion and landing. We believe this was purely mechanical and the high oil temperature was out of our control.