Step 1: Understand the task.
We must order five ideas $K, L, M, N, O$ into a coherent precis on disarmament.
Step 2: Find the opening idea.
$(L)$ states the nuclear age offers only annihilation or peaceful co-existence; this grand dilemma must frame everything, so it comes first.
Step 3: Find the response to the dilemma.
$(K)$ says disarmament gains overriding importance, the natural answer to the threat in $L$, so $L \rightarrow K$.
Step 4: Add the method.
$(M)$ explains that peace needs tolerance and reshaped policies, the how of disarmament, giving $L \rightarrow K \rightarrow M$.
Step 5: Add urgency, then authority.
$(N)$ stresses that delay is dangerous, injecting urgency, and $(O)$ names the United Nations as the responsible body, a fitting close.
Step 6: Assemble the chain.
The order is $L \rightarrow K \rightarrow M \rightarrow N \rightarrow O$, which is option A.
\[ \boxed{\text{A. } L \rightarrow K \rightarrow M \rightarrow N \rightarrow O} \]