waterbear answered this well. I'm going to try and answer it from a slightly different angle.
The basic question that is being asked is "if something is separated, then why doesn't it combine later?" The answer is that chemistry (and physics) doesn't work that way. Chemical and physical systems move towards their lowest energy state. They don't always do so quickly -- otherwise our bodies would simply decompose (i.e. "burn") into carbon dioxide, water and possibly soot -- but the direction is towards lower energy. Salt that is solid when not in water is in a lower energy state than salt in a liquid or gaseous form (at room temperature) so is why you see it as solid salt when not in water. When salt is exposed to water, the sodium and chloride in the salt that are attracted to each other when there is no water present become separated when in water because the sodium and chloride are separately more attracted to water than they are to each other (in some sense) and are in a lower energy state when separate as charged ions. This occurs because water is polar so has a positive charge on one side of the molecule and a negative charge on the other so the net result is that the water and the sodium and chloride all bounce around in a way that is at the lowest energy.
Sodium in water, as sodium ions (that is, charged atoms interacting with the polar water molecules), is at a very low energy state so it will pretty much not react with or combine with anything. It is very hard to precipitate anything with sodium in water without actually removing the water itself (i.e. without evaporating the water). So the hypochlorous acid that is in the water and the hypochlorite ions that are in the water do not react with nor combine with the sodium. When you add bleach or chlorinating liquid, this contains sodium hypochlorite and in water the sodium ion and the hypochlorite ion remain separated, just like sodium and chloride from salt dissolved in water. The principle is very similar -- pretty much anything initially combined with sodium (and potassium, to a slightly lesser extent) remain separated when in water.
Even with extra hydroxide in water, sodium hydroxide is still separate sodium ions and hydroxide ions whenever water is present. The salt cell does produce an excess of hydroxide ions, but they just remain as separate hydroxide ions and don't combine with the sodium. Also, when the chlorine (hypochlorous acid) gets used up either from breakdown from sunlight of from oxidation of organics, this produces an excess of hydrogen ions and these combine with the hydroxyl ions to form water so the net result of chlorine addition from a hypochlorite source of chlorine (such as bleach or chlorinating liquid or Cal-Hypo) or from creation in an SWG cell PLUS the later breakdown of chlorine is a net result that is pH neutral -- no excess of hydroxyl ions or hydrogen ions. (Technically, there is a small amount of extra hydroxyl ion in the hypochlorite sources of chlorine, but it is negligible except after months of usage).
Hope that helps in understanding.
Richard