The British conquest of the territory that would become Nigeria was a gradual process spanning several decades, and it was completed with the fall of the Sokoto Caliphate in 1903.
The Sokoto Caliphate, founded by Uthman dan Fodio following the Fulani Jihad of 1804, was the largest and most powerful state in what is now northern Nigeria. It controlled a vast territory encompassing most of Hausaland and beyond. The British, under Sir Frederick Lugard (High Commissioner of the Protectorate of Northern Nigeria), launched a military campaign against the Caliphate. The capital, Sokoto, fell to British forces in March 1903 when the last Sultan, Muhammad Attahiru I, was defeated and fled. His subsequent death at the Battle of Burmi in July 1903 effectively ended organised resistance.
While the Benin Kingdom was conquered by the British in the Punitive Expedition of 1897, this preceded the northern campaigns. The Kanem-Bornu Empire had already declined significantly before British arrival. Various Yoruba kingdoms had come under British influence through treaties and the end of the Yoruba civil wars in the 1890s, but the Yoruba territories were not the final piece of the conquest.
The defeat of the Sokoto Caliphate marked the completion of British territorial control over what became Nigeria, making it the correct answer.