It is slow to do the cough off at first. And that is totally ok, it is good to really get the feeling of the relaxation, removing all the air, and then the body automatically refilling. Also the fact you use the word gasp, tells me you are taking in way too much air and over supporting.
Whenever you live on small amount of air, and learn complete fold closure, it only takes a small amount of air to create big sound. As I said last post, making sure you close your vocal folds close first then begin support will really help you align that. Also will help you sing long phrases with out having to re tank. Act like you are about to say something then stop, hold that feeling and breath, you will notice the onset of sound. That is the basic feeling of closing, then sound.
As for time frame it takes, when you get things balanced, you can complete the exhale then inhale at the same time or quicker than a regular inhale only. I know it is quicker for me now, than just trying to inhale. I dont have to think about breathing, there are just certain spots in the piece that i know i will need to breath(obviously from being familiar with the piece). You really dont need any more air to sing than speaking, its just more compressed. I just add the quick little exhale and relaxation at the end of a selected phrase.
So for example i would say baaaabbbbyyyyyy the yyyy is compressed then, when I am ready to end the sound, I just release the folds but keep the compression. So sound stops, the rest of the air is quickly released(its only a tiny amount because you only need a small amount, and you have already expelled some from the phrase) then the body has no air, so it refills automatically. Plus during all this, you have that moment of relaxation, and your system gets to "reset". You can hear good opera singers do this all the time, Pavarotti for example. Even some of the power house contemporary singers do it, they just no how to either add it to their style, or hide it all together. I used pavarotti as an example because he is slightly obvious when he does it.
All of this blends in at the end of a phrase, and when the timing is down it takes less time than tanking. Also relieves constriction and helps you stay relaxed, so you are fresh for the next phrase.
As for breathing points right now on that song, I would handle them a little differently. However, i would focus on your balance first. Just remember, a little can make ALOT singing, and relaxation is golden.
Oh and I already said something about the soft palate in your thread about the soft palate