All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Neither Ibn Kathir in Al-Bidayah Wa al-Nihayah nor other historians mentioned that Ibn Az-Zubayr revolted against 'Uthman . None of the Companions revolted against 'Uthman . Rather, Ibn Kathir mentioned in Al-Bidayah Wa al-Nihayah that Ibn Az-Zubayr was one of those who defended 'Uthman when the unruly Kharijite besieged him. Ibn Kathir said about the story of the Kharijite's siege of 'Uthman : “A group of the sons of the Companions marched to him on the orders of their fathers. Among them were Al-Hasan and Al-Husayn, 'Abdullah ibn Az-Zubayr - who was in charge of the house (i.e. in charge of his defense) - and Abdullah ibn 'Amr and they began arguing on him and defending him so that no one of them could reach him.” [End of quote]
Whatever happened between them, the approach of Ahlus-Sunnah is to refrain from delving into what happened between the Companions during the days of fitnah (trial, tribulation), because there is no desired benefit from delving into that. Rather, it would lead to attributing any deliberate mistake to any of the Sahaabah in a matter in which they may not have been mistaken in reality, or they may have been mistaken but were excused. They tried their best to decide correctly in what happened between them, there are two rewards for those who were right, and one reward for those who erred due to their ijtihad, and their mistakes are excused and forgiven by Allah the Almighty.
Allah knows best.