That was a nuclear engine for a cruise missile, not a nuclear warhead.

Technically correct.

Now, earnest question: what happens to the nuclear engine fuel at end of flight? While it certainly won't become critical it's likely the only part it stops being is "engine fuel".

(Of course the question above is irrelevant the moment the missile is fitted with a nuclear warhead)

Of course this nuclear cruise missile is horribly dangerous with or without a nuclear warhead.

However, testing it says nothing about whether the nuclear warheads still work, which is the nuclear testing in question.

Nuclear missile is effectively a flying scramjet nuclear reactor heating air passing through it via radioactive decay, not a nuke.