It's in the dialogue that it's taken the Klingons months to repair their ship to almost functionality. Voq was showing all the ship debris points they'd scavenged from on that holomap. Did you notice how intricate the lines were? Searching through salvage and enacting repairs takes time. It wasn't that they just needed one part only and stubbornly waited around doing nothing for six months until Voq changed his mind. There's no evidence that the dilithium processor was anything more than a standard part for dilithium warp drives. Yes the part was valuable and critical to Voq's ship, but only because they hadn't been able to find a salvagable one elsewhere in the debris field.