A Center Oceanic Controller trainee reported they used less than required non radar separation between an aircraft descending/turning unexpectedly and another enroute aircraft.

Narrative

Working ATOP in sector XX; Day 0 at XA50 received CPDLC message from Aircraft X; [priority handling] request HF Voice contact. I called ZZZ Radio and requested phone patch; pilot advised that they had a #2 engine failure; and were descending to 24000 ft. and turning right; requesting direct ZZZZ. Pilot advised that they would use CPDLC as primary communication. I noticed Aircraft Y could potentially be in conflict at 33000 ft. and at XA53 changed his route from direct ZZZZZ to direct ZZZZZ1 to route him further south of Aircraft X and issued traffic alert '[priority aircraft] approximately 45 miles 3 o'clock FL340 descending to FL240'. Once I issued direct ZZZZ and block altitude 24000 to 26000 ft. ATOP showed conflict between the two aircraft without the eligibility for reduced 30 or 50 miles separation. Once aircraft profiles were updated and conflict was resolved reduced the Automatic Dependent Surveillance (ADS) periodic rate to 64 seconds in Aeronautical Facility Notification (AFN) window. [I suggest] Non Radar [problem] response refresher.