It always means both. You can substitute the word 'when' for the
phrase 'as soon as' and both meanings will be preserved. You can
also substitute 'just after'. As soon as the clock strikes twelve,
the trumpets will sound. When the clock strikes twelve, the
trumpets will sound. Just after the clock strikes twelve, the
trumpets will sound. The phrase 'as soon as' refers to the moment
that something will occur. The rest of the sentence tells us what
happens 'just after'.