Can I omit the "neither" in a neither/nor construction? [duplicate]
The neither is required.
Neither ... nor acts as a negation which as you correctly note would otherwise require "did not" (past tense negation in English is a bit verbose).
You could also have written
Something was awry but Alice and Bob did not know what.
(Note that the "neither...nor" turns into "and" here, one of those rare instances of linguistic grammar conforming to Boolean logic).