Quirk in NF timeout rule?

[Tee]

Could you please explain under NFHS- at what point after a made basket can the defensive team be granted a time out?

Ball drops out of the basket-

Offensive player grabs ball and heads for the baseline-

Offensive player is set up to throw ball inbounds-

I have always used the "at the disposal of the thrower" to mean a reasonable amount of time to recover the ball- at that point I begin my count. I guess if I'm counting that means no timeout?

Sometimes the thrower chooses not to pick up the ball, hoping I will delay my count until he has the ball in his hand- can a timeout be granted to the defensive team then? I will be counting...

Another complication- the thrower throws the ball to me (as they do occasionally) does this affect the count? I see many officials begin the count at that point- thus effectivly allowing an extra second or so to the thrower.