The operative phrase here is "at the disposal of". We have an entirely separate thread going on this exact point, and there is disagreement there.
Generally, there are specific game situations in which a good official will anticipate that a coach is likely to request a timeout following a score. We know what they are - the other team is making a run, it's late in a close game and the other team scores, etc. In those cases, I usually turn my head toward that bench quickly to see if the coach is signaling and if he is before the other team picks up the ball, or has the opportunity to do so (my interpretation of "disposal"), I will grant the timeout, even if my "whistle-lag" takes a moment.
However, in times other than that, we are not anticipating the request (after all, we cannot check the benches after every score), so I have to make a judgement as to whether the request came in time or not.
My suggestion is to immediately shout "timeout" at the top of your lungs while making either a T sign or a 30 second gesture with your hands as soon as the ball clears the net. Even in a noisy gym, that's the most likely way to get our attention.
Of course, whichever way I decide, I'm always right.
