Is there a longer duration between me pushing the button to take a photo and the photo being taken depending on what quality I select?

Keep something in mind...

There are two latencies, and I think the latency you're worried about isn't affected by the storage media or the quality level.

The latency that's affected by the quality level and the speed of the storage flash is the amount of time between pictures. For example, how long after you've taken a picture can you take another one. I don't think that's what you're asking.

What you're asking is, how can you reduce the amount of time between the time you press the button and the time it actually goes "click". That's controlled by something completely different.

The things that are killing you there are the auto-focus, auto-exposure control, and auto-whitebalance. The camera actually performs all of those functions when you press the button, and they all take time, especially the focus.

If you have a camera that can be set to manual focus, manual exposure control, and manual white balance, then your latency between "press" and "click" is almost instant.

Another way around it, that some cameras do (my Canon does this) is that the shutter button is two-stage. You press gently for it to do the exposure/whitebal/focus procedure, then press it further down all the way to actually take the exposure. This allows you to aim and focus and set up your shot (say, focusing on the birthday cake and candles), and then allow you to click the shutter release instantly at the moment the kid blows out the candles. (Or swings the bat, or jumps the pole vault bar, or whatever.)