LONDON — It has consumed more than eight years of maneuvering, obstruction and a widow’s dogged legal campaign, fought often on a shoestring. But finally, on Tuesday, a public inquiry is set to begin its quest for an answer to the question that has driven the whole process: Why did Alexander V. Litvinenko have to die? On Nov. 1, 2006, Mr. Litvinenko, a former officer of the Soviet K.G.B. in self-exile in London and a vocal critic of President Vladimir V. Putin, sipped tea from a poisoned pot, took sick and died 22 days later. Only after his death did...