SoCal_Dad is getting real close to answering the question asked as I read it. I believe we have some confusion on this topic. Hopefully Bretman can shed more light after my post.
There are a couple or more of rules being tested here. The first is the rule pertaining to the batter. The batter is required to get her signal and assume her position in the batter's box prior to the umpire awarding the "play ball" signal and command. That's when the pitcher can bring her hands together and starts the 10-second count. A pitcher cannot assume her position on the pitching plate and bring her hands together just because she has the ball. She has to wait till the umpire puts the game back into play. NOW, once she recieves the order or signal from the umpire, she CAN bring her hands together, separate, then complete her pitch regardless of the batter's stance. The umpire will then award either a strike or ball on the pitch thrown.
The other rule that needs explanation deals with the time limit on the batter. Many sanctions say 20 seconds but there is some confusion on when this count should start. First of all, the batter requests time out. The umpire awards it if he feels it needed. Time out to the batter is not normally awarded once the pitcher's hands come together unless there is a situation of safety involved such as an errant ball on the field. Once the pitcher brings her hands together, that engages play. The batter can ask for time but normally it is never awarded. If she backs out of the box and the pitcher completes her pitch for a strike, the strike call will be awarded. Yes, pitchers have been known to hold their pose with both hands with the ball together for an entire 10 count. It can be annoying to a batter. Still, it is a part of the game. I teach my hitters to assume a position in the box and remain relaxed with hands loose until the pitcher separates her hands. That's the sign she's going to deliver a pitch.
Last, the rule about the pitcher and the time she has to throw the next pitch. Most sanctions say the pitcher has 20 seconds from the time she possesses the ball in the pitcher's circle till she is required to throw her next pitch. This is true. She can have that time within the circle to safely compose herself, wipe off the ball, fix her hair, pick her nose, (although still required to wipe her hands off before gripping the ball), before taking her stance on the pitching plate. Her stance on the plate indicates to the umpires she is ready. She then pauses to take or simulate taking a sign. her next move would be to bring her hands with the ball together but only if the umpire has indicated that the game is now in play. She can request time and if granted step back, but normally not awarded once her hands come together. Once they come together she needs to complete her pitch