How was the next leader of the Uchiha clan chosen?
Right before Madara left his own clan, Konoha was founded as a place where Uchiha and Senju could co-exist peacefully. 

After Madara's death, there was no true leader for the Uchiha clan. All of them were under one leadership of Hokage along with rest of the citizens of Konoha. Hashirama was elected as the first Hokage and he was later succeeded by Tobirama.