A Center Controller reported a cargo carrier had to descend unexpectedly due to pressurization failure.

Narrative

Aircraft X landing ZZZ checked in normally at FL310. After a few minutes he requested an emergency descent to 10;000 ft due to cabin pressure loss. I turned him 20 degrees to the right to miss Aircraft Y; a ZZZ1 arrival who was descending to cross ZZZZZ at FL240. I had already shipped Aircraft Y to the next sector prior to Aircraft X stating they needed an emergency descent. The pilot of Aircraft X read back FL240 and 20 degrees. I thought Aircraft X had read back the right part of the turn and was going to descend very rapidly due to cabin pressure loss and so I hopped on the line with sector XX to coordinate lower as the base of my altitude stratum was FL240. They gave 16;000 and I prompted them to give me an altimeter since I was only high altitude stratum at this sector. I came off the line and by the time I was finished coordinating I noticed that Aircraft X had started to turn left towards Aircraft Y. I corrected the turn and issued the traffic and told Aircraft X to expedite the turn. They called the traffic in sight. Every read back the pilot gave after they told me they needed an emergency descent to 10;000. I was having trouble understanding probably because the pilots had their oxygen masks donned and the sound quality was very poor. The pilot leveled at FL240 for a while before descending to 16;000 for a while so I had asked a couple times to verify they still needed lower since they were not descending how I had anticipated. The strong westerly winds at FL330 and above I thought were contributing to the lack of turn and so there were multiple factors as to why I thought there was a delay in the turn. If I would have verified that the pilot did indeed read back right turn 20 degrees I would have been able to save it but I knew I didn't hear 20 degrees left and the pilot turned the exact opposite direction I told him to.