I have corresponded with Seachem on the issue of Prime so perhaps this rather simple explanation may help. I told them I was not a chemist and didn't want to get bogged down in convoluted equations, so they kept it simple for me.
Prime "detoxifies" ammonia, nitrite and nitrate by binding them somehow. In the case of ammonia, it becomes ammonium, which is basically harmless. Nitrosomonas bacteria will use ammonia or ammonium, whichever is present, and live plants the same [plants prefer ammonium as their nitrogen source, and have the ability to take up ammonia and convert it into ammonium]. As for nitrite and nitrate, Seachem stated they are not really certain how this works themselves, but there is some sort of binding. And here again, bacteria can still take them up.
Second, Prime is effective for a limited period. Seachem suggested 24-48 hours, with 36 hours perhaps being safer. The idea behind using Prime if you have ammonia, nitrite or nitrate in the source water (or in the aquarium as during the initial cycle) is that it will render all three harmless for 24-36 hours. If the substances are still present after this period, they will "unbind" so to speak, and become toxic. But in a normal balanced aquarium the plants and/or bacteria should be able to handle the ammonia/nitrite/nitrate introduced in the source water at a water change by the time Prime wears out. In new cycling tanks, additional water changes using Prime perhaps daily will be needed until the bacteria is established sufficiently to handle the ammonia and nitrite. But here, if live plants are present in sufficient quantity with some fast grtowers, they will be able to deal with the ammonia [with a minimal fish stocking obviously] and nitrite is not a byproduct with plants, which is why live plants in new tanks are so important.
The chlorine and chloramine is a bit different; it remains detoxified.
A final few words on the ammonia/ammonium. The pH does affect this. At an acidic pH, below 7, the ammonia naturally occurring from fish respiration, breakdown of organics, etc will be in the "safe" ammonium form. Plants and bacteria will take it up. But the fish will not be affected because it is ammonium. But in a basic pH, above 7, it remains ammonia [except when Prime or a similar product is present]. So if Prime is used and the pH is above 7, the ammonia is bound into ammonium for 24-36 hours, after which it will "unbind" back into toxic ammonia in the basic pH.