The watershed is the time limit after which more controversial subjects, bad language, etc, can be shown on TV in some countries, so if it's after the watershed, then discussions can be freer, franker and more controversial.
If something carries the day, it wins a battle (the sense is that the battle has been long and could have gone either way) or competition for supremacy.
(Irish) This idiom is used when something is obvious because of the day that it occurs: traffic, for example would be busy around a football stadium on game day, given the day that's in it. On any other day the traffic would be unexplainable, but because its game day its obvious why there is traffic.
If people say that there's no time like the present , they believe that it is far better to do something now than to leave it for later, in which case it might never get done.