On a solo world, time in the nether seems to stop time in the overworld. Will the phantom spawning timing still continue? If I spend 2 hours RT in nether or equivalent to 6 overworld days, is the phantom spawning timer set to 6 days plus what it was or does this timing stop while in nether?
The Wiki doesn't seem to make any mention of this but it appears that time spent in the Nether does count (for example, it is mentioned in this Reddit thread); while it is true that even the spawn chunks are unloaded when you leave the Overworld, thus no processing occurs, game time continues to progress as usual - I've gone to the Nether while it was day and it was night when I returned and my mod's "inventory stats" continued to show the passing of days (based on day/night cycles, not overall time, which is a separate variable which is unaffected by sleeping or time commands).
Of course, this is in 1.6.4 but I don't see why this would have changed since then; the same also applies to the End, and the game may very well simply check for when the time since you last slept reaches 1 hour, and there is a statistic which tracks this, "Since Last Rest" (listed near the bottom), which is used to control the spawning of phantoms.
The day/night cycle continues if you are in the nether or overworld or end. If you have a villager in the nether or end and a bed available they will still sleep on the day/night cycle as if they were in the overworld.
If you spend the equivalent of 6 minecraft days in the nether, then you will spawn phantoms in the overworld equivalent to staying up 6 days/nights. It sounds like what you are talking about things like furnaces not smelting or plants not growing while you are in the nether. That is because those are conditional on player location. A player has to be loading the chunks for those things to progress.