Why wartime royal succession would preserve the Crown

Would Britain become a republic if the royal family were killed deep into the line of succession during the Second World War?

No; a remote successor would take the throne because wartime politics favored continuity of the Crown rather than abolition.

The Crown could continue as a constitutional institution even after the deaths of its most immediate successors. The expectation of continued monarchy rests on Churchill's wartime leadership and the absence of a senior republican constituency in government. Those conditions would favor finding a successor through the existing institution. A distant relative could therefore inherit without turning the emergency into a change of constitutional system.